Recent Progress In Many-Body Theories: Proceedings of the 14th International Conference Barcelona, Spain 16 - 20 July 2007 (Advances in Quantum Many-Body Theory)

RECENT PROGRESS Lfi MANY-BODY THEORIES Series on Advances in Quantum Many-Body Theory Edited by R. F. Bishop, C. E. ...

Author: Jordi Boronat | Greogory E Astrakharchik | Ferran Mazzanti

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RECENT PROGRESS Lfi

MANY-BODY THEORIES

Series on Advances in Quantum Many-Body Theory Edited by R. F. Bishop, C. E. Campell, J. W. Clark and S. Fantoni (International Advisory Committee for the Series of International Conferences on Recent Progress in Many-Body Theories)

Published Vol. 1:

Proceedings of the Ninth International Conference on Recent Progress in Many-Body Theories Edited by D. Neilson and R. F. Bishop

Vol. 3:

Proceedings of the Tenth International Conference on Recent Progress in Many-Body Theories Edited by R. F. Bishop, K. A. Gernoth, N. R. Walet and Y. Xian

Vol. 4:

Microscopic Approaches to Quantum Liquids in Confined Geometries E. Krotscheck and J. Navarro

Vol. 5:

150 Years of Quantum Many-Body Theory A Festschrift in Honour of the 65th Birthdays of John W Clark, Alpo J Kallio, Manfred L Ristig and Sergio Rosati Raymond F. Bishop, Klaus A. Gernoth and Niels R. Walet

Vol. 6:

Proceedings of the Eleventh International Conference on Recent Progress in Many-Body Theories Edited by Raymond F. Bishop, Tobias Brandes, Klaus A. Gernoth, Niels R. Walet and Yang Xian

Vol. 7:

Introduction to Modern Methods of Quantum Many-Body Theory and Their Applications Adelchi Fabrocini, Stefano Fantoni and Eckhard Krotscheck

Vol. 9:

Proceedings of the Twelfth International Conference on Recent Progress in Many-Body Theories Edited by J. A. Carlson and G. Ortiz

Vol. 10: Proceedings of the Thirteenth International Conference on Recent Progress in Many-Body Theories Edited by S. Hernández and H. Cataldo Vol. 11: Proceedings of the Fourteenth International Conference on Recent Progress in Many-Body Theories Edited by J. Boronat, G. Astrakharchik and F. Mazzanti

Forthcoming Vol. 2:

Microscopic Approaches to the Structure of Light Nuclei Edited by R. F. Bishop and N. R. Walet

Vol. 8:

Pairing in Fermionic Systems: Basic Concepts and Modern Applications Edited by S. Armen, M. Alford and J. W. Clark

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Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Series on Advances in Quantum Many-Body Theories RECENT PROGRESS IN MANY-BODY THEORIES Proceedings of the 14th International Conference Copyright © 2008 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN-13 978-981-277-987-8 ISBN-10 981-277-987-6

Printed in Singapore.

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FOREWORD The Fourteenth International Conference on Recent Progress in Many-Body Theories (RPMBT-14) was held at the Technical University of Catalonia (UPC), Barcelona, Spain, 16-20 July 2007. The present volume contains most of the invited talks plus a selection of excellent poster presentations. This conference series is now firmly established as one of the premier series of international meetings in the field of Many-Body physics. The first official RPMBT meeting was held in Trieste in 1978, in response to several precursor meetings that accentuated the need for a continuing series. The most important of these, which can be regarded as RPMBT-0, is the 1972 conference on The Nuclear Many-Body Problem organized by F. Calogero and C. Cioffi degli Atti in Rome. Additionally, there were two very significant workshops held in 1975 and 1977 at the University of Illinois, Urbana, with Vijay Pandharipande as the chief organizer. Later conferences in the series have been the 1981 RPMBT-2 meeting in Oaxtepec, Mexico; the 1983 RMPBT-3 meeting in Altenberg, Germany; the 1985 RPMBT-4 meeting in San Francisco, USA; the 1987 RPMBT-5 meeting in Oulu, Finland; the 1989 RPMBT6 meeting in Arad, Israel; the 1991 RPMBT-7 meeting in Minneapolis, USA; the 1994 RPMBT-8 meeting in Schloss Seggau, Styria, Austria; the 1997 RPMBT-9 meeting in Sydney, Australia; the 1999 RPMBT-10 meeting in Seattle, USA; the 2001 RPMBT-11 meeting in Manchester, UK; the 2004 RPMBT-12 meeting in Santa Fe, USA; the 2005 RPMBT-13 meeting at Buenos Aires, Argentina, and the present 2007 meeting in Barcelona, Spain. Highlights and a more detailed history of past meetings can be found in earlier volumes of this series. This conference series is also responsible for awarding the prestigious Eugene Feenberg Memorial Medal in Many-Body Physics. This Medal, first presented in 1985, is designated for work that is firmly established and that can be demonstrated to have significantly advanced the field of many-body physics. The work considered can be accumulative contributions sustained over time, or a single important contribution. In appropriate cases, the award can be shared by as many as three people for a single body of work. More details on the Feenberg Medal and its nomination process can be found in the Conference Series official website http://www.qmbt.org/Feenberg/index.php?doc=Feenberg. Past recipients have included David Pines (1985), John W. Clark (1987), Malvin H. Kalos (1989), Walter Kohn (1991), David M. Ceperley (1994), Lev P. Pitaevskii (1997), Anthony J. Leggett (1999), Philippe Noziéres (2001), Spartak T. Belyaev and Lev P. Gor’kov (2004), and Raymond F. Bishop and Hermann G. K¨ ummel

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(2005). Professors Kohn and Leggett received the Nobel Prize in 1998 and 2003 respectively. We are pleased that the Feenberg Medal was awarded at this conference to Professors Stefano Fantoni and Eckhard Krotscheck, “for their leading role in the development and extensive applications of the correlated basis function method, including the advance of Fermi hypernetted chain theory, thereby providing an accurate, quantitative, microscopic description of strongly-interacting quantum many-body systems, especially for finite atomic nuclei and inhomogeneous quantum fluids”. In addition to their outstanding research achievments, both have been inspirational models for a generation of many-body theorists. The presentation was made by Jordi Boronat, Chair of the Feenberg Medal Selection Committee in a special Award Session held in Barcelona’s world famous Science Museum CosmoCaixa. The text of the tribute, as well as the responses of the Medal recipients are included in this volume. Another highlight of this conference is the presentation of the inaugural Hermann K¨ ummel Early Achievement Award in Many-Body Physics to Dr. Frank Verstraete of Universit Wien, Austria, in recognition of his pioneering work on quantum information and entanglement. This award was established by the International Advisory Committee of the Conference Series to recognize outstanding published work done within six years of receiving the doctorate degree. The award honors Prof. K¨ ummel’s long and distinguished career as a leader in the field of many-body physics and as a mentor of younger generations of many-body physicists. More details on the K¨ ummel award can again be found at http://www.qmbt.org/Kuemmel/index.php?doc=KuemmelAward. The quality of nominees for this award was so outstanding that the Selection Committee recommended that Honorable Mentions be given to Dr. Gregory E. Astrakharchik of the Technical University of Catalonia, Barcelona, Spain, for his calculation of the BEC-BCS crossover in dilute Fermi gases, and to Dr. Robert Zillich of Johannes Kepler Universit¨ at, Linz, Austria, for his quantum Monte Carlo simulation of strongly correlated quantum fluids. The presentation was made by Susana Hernandez, Chair of the K¨ ummel Award Selection Committee, in the special Award Session. The current conference maintains the tradition of covering the entire spectrum of theoretical tools developed to tackle important and current quantum many-body problems, with the aim of fostering the exchange of ideas and techniques among physicists working in diverse subfields of physics. The highlights of the conference included state-of-the-art contributions on the dynamics and rotation of ultra-cold quantum gases, BEC-BCS cross-over, quantum liquid and solids, correlated electron systems and superconductivity, correlated nuclear systems and nuclear astrophysics, quantum computations and quantum Monte Carlo simulations. The conference continues to demonstrate the vitality and the fundamental importance of many-body theories, techniques, and applications in understanding diverse and novel phenonomena at the cutting-edge of physics.

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We thank the Program Advisory Committee for recommending excellent topics and great speakers for the Conference. We are very much indebted to each member of the Local Organizing Committee: Jordi Boronat, Artur Polls, Jes´ us Navarro, Joaquim Casulleras, Ferran Mazzanti, Muntsa Guilleumas and Gregory Astrakharchik, for their tireless labor and attention to details, which made this meeting productive and memorable. Above all, we are all in awe of Jordi’s singular devotion and energy, which made this conference possible. Siu A. Chin Chair, International Advisory Committee for the Series of International Conferences on Recent Progress in Many-Body Theories College Station, U.S.A.

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ORGANIZING COMMITTEES SERIES EDITORIAL BOARD for the Series on Advances in Quantum Many-Body Theory

R. F. Bishop (Chairman) C. E. Campbell J. W. Clark S. Fantoni

– – – –

UMIST, Manchester, UK University of Minnesota, USA Washington University, St. Louis, USA SISSA, Trieste, Italy

INTERNATIONAL ADVISORY COMMITTEE for the Series of International Conferences on Recent Progress in Many-Body Theories

Siu A. Chin (Chairman) Charles E. Campbell (Treasurer) Hermann K¨ ummel (Hon. President) Raymond F. Bishop Joe A. Carlson John W. Clark Peter Fulde

– – – – – – –

Susana Hern´ andez Eckhard Krotscheck Claire Lhuillier Allan MacDonald Efstratios Manousakis David Neilson Gerardo Ortiz Artur Polls Mikko Saarela Masahito Ueda

– – – – – – – – – –

Texas A & M University, USA University of Minnesota, USA Ruhr-Universit¨ at Bochum, Germany University of Manchester , UK Los Alamos National Laboratory, USA Washington University, St. Louis, USA Max-Planck-Institut f¨ ur Komplexer Systeme, Dresden, Germany Universidad de Buenos Aires, Argentina Johannes Kepler Universit¨ at Linz, Austria Université Pierre et Marie Curie, France Indiana University, USA Florida State University, USA University of Camerino, Italy Indiana University, USA Universitat de Barcelona, Spain University of Oulu, Finland Tokyo Institute of Technology, Japan

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PROGRAMME COMMITTEE FOR THE FOURTEENTH CONFERENCE Marcello Baldo Manuel Barranco Raymond Bishop Enrique Buend´ıa Charles E. Campbell Joe Carlson Ignacio Cirac John W. Clark Jorge Dukelsky Stefano Giorgini Susana Hern´ andez Morten Hjorth-Jensen Eckhard Krotscheck David Neilson Lubos Mitas Gerardo Ortiz Luciano Reatto Mikko Saarela Kevin Schmidt Masahito Ueda

– – – – – – – – – – – – – – – – – – – –

INFN, Italy Universitat de Barcelona ,Spain University of Manchester, UK Universidad de Granada, Spain University of Minessota, USA LANL, USA MPQ, Germany Washington University, St. Louis, USA CSIC , Spain University of Trento ,Italy Universidad de Buenos Aires, Argentina University of Oslo, Norway Johannes Kepler Universit¨ at Linz, Austria University of Camerino, Italy North Carolina State University, USA Indiana University, USA University of Milan, Italy University of Oulu, Finland Arizona State University, USA Tokyo Institute of Technology, Japan

LOCAL ORGANIZING COMMITTEE FOR THE FOURTEENTH CONFERENCE Jordi Boronat (Chairman) Gregory E. Astrakharchik Joaquim Casulleras Muntsa Guilleumas Ferran Mazzanti Jes´ us Navarro Artur Polls

– – – – – – –

Universitat Politècnica de Catalunya Universitat Politècnica de Catalunya Universitat Politècnica de Catalunya Universitat de Barcelona Universitat Politècnica de Catalunya Institut de F´ısica Corpuscular (CSIC-UV) Universitat de Barcelona

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PREFACE The Fourteenth International Conference on Recent Progress in Many-Body Theories (RPMBT14) was held at the Technical University of Catalonia (UPC), Barcelona, Spain over the period 16-20 July, 2007. The sessions of the meeting were organized according to a list of topics proposed by the Programme Committee spanning traditional fields of the Conference Series, such as Quantum Fluids and Solids and Nuclear Physics, and newer fields like Quantum Computation or Cold Quantum Gases, all of them with the common link of Quantum Many-Body Physics. During the Conference, some 45 invited papers were presented orally and approximately 40 contributed papers were presented as posters. All the speakers were requested to submit a paper for this Proceedings Volume and around ninety per cent of them submitted the paper by the tight deadline imposed. Moreover, the Editors made a selection among the poster contributions and offered to some of the authors the opportunity of including their contributions in the present Volume. Some of the papers are quite interdisciplinary and therefore difficult to classify into a numbered list of topics. Nevertheless, we have decided to group the papers according to the topic session in which they were classified by the Programme and Organizing Committees. The Editors suggest to consider this classification as a guide to the readers, and recommend that they go through the entire Volume to get a deeper insight on the recent progress in the field. We would like to thank all the people who have made it possible for this Volume to arrive to your hands. First, we warmly thank all the authors contributing to this Volume for their accurate work and close collaboration, and for amending any format problem that emerged during the edition. Second, we would like to thank Siu Chin, Chair of the International Advisory Committee of the Conference Series, for enriching this Volume with a clever preface, to John Clark for his warm remembrance of Eugene Feenberg at the centenary of his birth. We are also grateful to Gerardo Ortiz, Artur Polls and Mikko Saarela for their Laudations of the 2007 Kuemmel Award and Feenberg Medal winners. Finally, we are indebted to the rest of the members of the Organizing Committee, Joaquim Casulleras, Muntsa Guilleumas, Jes´ us Navarro and Artur Polls, for their invaluable advices and help in the organization of the RPMBT14 Conference. Jordi Boronat Gregory E. Astrakharchik Ferran Mazzanti

Barcelona, Spain 5 January 2008

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Group picture

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CONTENTS Foreword

v

Organizing Committees

ix

Preface

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FEENBERG MEDAL AND KUEMMEL AWARD PRESENTATIONS

1

The legacy of Eugene Feenberg at the centenary of his birth J. W. Clark

3

Stefano Fantoni: Feenberg Medalist 2007: Microscopic Many-Body Theory of Strongly Correlated Systems A. Polls

11

Eckhard Krotscheck: Feenberg Medalist 2007: Microscopic ManyBody Theory of Quantum Fluids M. Saarela

16

Frank Verstraete: Hermann Kuemmel Award 2007 G. Ortiz

20

Quantum Monte Carlo calculations for nuclei and nuclear matter S. Fantoni, S. Gandolfi, F. Pederiva, and K. E. Schmidt

23

Static and Dynamic Many-Body Correlations E. Krotscheck and C. E. Campbell

39

Entanglement in many-body quantum physics F. Verstraete

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COLD BOSE AND FERMI GASES

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New states of quantum matter G. Baym

65

Stationary Josephson effect in the BCS–BEC Crossover A. Spuntarelli, P. Pieri, and G. C. Strinati

75

Ultra-cold dipolar gases C. Menotti and M. Lewenstein

79

Crystalline phase of strongly interacting Fermi mixtures D. D. Petrov, G. E. Astrakharchik, D. Papoular, C. Salomon, and G. V. Shlyapnikov

94

Localization and glassiness of bosonic mixtures in optical lattices T. Roscilde, B. Horstmann, and J. I. Cirac

106

Scattering of a sound wave on a vortex in Bose–Einstein condensates P. Capuzzi, F. Federici, and M. P. Tosi

111

Static properties of a system of Bose hard rods in one dimension F. Mazzanti, G. E. Astrakharchik, J. Boronat, and J. Casulleras

116

A scenario for studying off-axis vortices in Bose–Einstein condensates D. M. Jezek, H. M. Cataldo, and P. Capuzzi

120

NUCLEAR AND SUBNUCLEAR PHYSICS

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Strangeness nuclear physics A. Ramos

127

Many-body methods for nuclear systems at subnuclear densities A. Sedrakian and J. W. Clark

138

Correlations as a function of nucleon asymmetry: The lure of dripline physics W. H. Dickhoff Fermi hypernetted chain description of doubly closed shell nuclei F. Arias de Saavedra, C. Bisconti, and G. Co’

148

152

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Many-body challenges in nuclear-astrophysics G. Mart´ınez-Pinedo

156

Coupled-cluster approach to an ab-initio description of nuclei D. J. Dean, G. Hagen, M. Hjorth-Jensen, and T. Papenbrock

168

Developing New Many-Body Approaches for No-Core Shell Model Calculations B. R. Barrett, A. F. Lisetskiy, P. Navr` atil, I. Stetcu, and J. P. Vary

172

Applications of in-medium chiral dynamics to nuclear structure P. Finelli

176

Variational Calculations of the Equation of State of Nuclear Matter M. Takano, H. Kanzawa, K. Oyamatsu, and K. Sumiyoshi

181

Refinement of the variational method with approximate energy expressions by taking into account 4-body cluster terms K. Tanaka and M. Takano

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COMPUTATIONAL QUANTUM MANY-BODY

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Nodal properties of fermion wave functions L. Mitas and M. Bajdich

193

Simulating rotating BEC: Vortices formation and over-critical rotations S. A. Chin

203

Polarizability in quantum dots via correlated quantum Monte Carlo L. Colletti, F. Pederiva, E. Lipparini, and C. J. Umrigar

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Progress in Coupled Electron-Ion Monte Carlo Simulations of High-Pressure Hydrogen C. Pierleoni, K. T. Delaney, M. A. Morales, D. M. Ceperley, and M. Holzmann

217

PHASE TRANSITIONS

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Quantum phase transitions on percolating lattices T. Vojta and J. A. Hoyos

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Ground-state properties of a homogeneous 2D system of Bosons with dipolar interactions G. E. Astrakharchik, J. Boronat, J. Casulleras, I. L. Kurbakov, and Yu. E. Lozovik Liquid-solid transition in Bose systems at T = 0 K: Analytic results about the ground state wave function E. Vitali, D. E. Galli, and L. Reatto The exact renormalization group and pairing in many-fermion systems N. R. Walet

245

251

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The spin-1/2 and spin-1 quantum J1 –J10 –J2 Heisenberg models on the square lattice R. F. Bishop, P. H. Y. Li, R. Darradi, and J. Richter

265

Liquid-gas phase transition in nuclear matter within a correlated approach A. Rios, A. Polls, A. Ramos, and H. M¨ uther

275

QUANTUM LIQUIDS AND SOLIDS

279

Small clusters of para-hydrogen R. Guardiola and J. Navarro

281

Adhesive forces on helium in nontrivial geometries E. S. Hern´ andez, A. Hernando, R. Mayol, and M. Pi

291

Rotational Spectra in Helium-4 Clusters and Droplets: Size Dependence and Rotational Linewidth R. E. Zillich and K. B. Whaley

295

Microscopic studies of solid 4 He with path integral projector Monte Carlo M. Rossi, R. Rota, E. Vitali, D. E. Galli, and L. Reatto

300

Liquid 4 He inside (10,10) carbon nanotubes M. C. Gordillo, J. Boronat and J. Casulleras

312

Spatial microstructure of fcc quantum crystals M. J. Harrison and K. A. Gernoth

316

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STRONGLY CORRELATED ELECTRONS

321

Nucleation of vortices in superconductors in confined geometries W. M. Wu, M. B. Sobnack, and F. V. Kusmartsev

323

The correlated density and the Bernoulli potential in superconductors K. Morawetz, P. Lipavský, and J. Kol´ aˇcek

332

Electron Correlations in Solids: From High-Temperature Superconductivity to Half-Metallic Ferromagnetism E. Arrigoni, L. Chioncel, H. Allmaier, M. Aichhorn, and W. Hanke

336

Excitons and polaritons in an optical lattice for cold-atoms within a cavity H. Zoubi and H. Ritsch

346

ATOMS AND MOLECULES

351

Fixed-Node Quantum Monte Carlo for Chemistry M. Caffarel and A. Ram´ırez-Sol´ıs

353

Quantum Monte Carlo for the electronic structure of atomic systems A. Sarsa, E. Buend´ıa, F. J. G´ alvez, and P. Maldonado

364

Hierarchical method for the dynamics of metal clusters in contact with an environment G. Bousquet, P. M. Dinh, J. Messud, E. Suraud, M. Baer, F. Fehrer, and P.-G. Reinhard Population transfer processes: From atoms to clusters and Bose–Einstein condensate V. O. Nesterenko, F. F. de Souza Cruz, E. L. Lapolli, and P.-G. Reinhard

374

379

QUANTUM COMPUTATION

385

Generalized entanglement in static and dynamic quantum phase transitions S. Deng, L. Viola, and G. Ortiz

387

Entanglement percolation in quantum networks: How to establish large distance quantum correlations? A. Ac´ın, M. Lewenstein, and J. I. Cirac

398

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NEW FRONTIERS

409

Phonon-roton excitations and quantum phase transitions in liquid 4 He in nanoporous media H. R. Glyde, J. V. Pearce, J. Bossy, and H. Schober

411

Topological quantum order: A new paradigm in the physics of matter Z. Nussinov and G. Ortiz

423

Thermal rectification in one-dimensional chains N. Zeng and J.-S. Wang

431

Author Index

435

Subject Index

437

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FEENBERG MEDAL AND KUEMMEL AWARD PRESENTATIONS

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THE LEGACY OF EUGENE FEENBERG AT THE CENTENARY OF HIS BIRTH JOHN W. CLARK Department of Physics, Washington University, St. Louis, Missouri 63130, USA ∗ E-mail: [email protected] Eugene Feenberg’s brilliant career in theoretical physics is reviewed, commemorating his vital role in the development of microscopic quantum many-body theory. The examples of his life and work continue to exert a profound influence on the character of the field, as reflected in the International Conferences on Recent Progress in Many-Body Theories. Keywords: Eugene Feenberg; history of physics; many-body theory.

1. A Guiding Theme Eugene Feenberg (1906-1977) emerges in the historical records of twentieth-century science as a leading pioneer in the application of quantum mechanics to nuclei and superfluid helium. In seeking an understanding of the behavior of these systems, he was not content with phenomenological descriptions or oversimplified models made popular by their tractability. Rather, his major contributions stemmed from a continuing quest (almost in his own words) for — Quantitative microscopic prediction of the observable properties of strongly interacting quantum many-body systems under realistic conditions of interaction, density, and temperature. This is often referred to as ab-initio theory (although the term has seen much abuse in recent years). Before the mid-1950’s, the most prominent theorists shared the attitude that such a goal was unattainable, either in principle or due to insufficient computational resources. It is a tribute to Feenberg’s vision that the guiding theme of his life work, expressed so visibly in this conference series on Recent Progress in ManyBody Theories (RPMBT), steadily gained ascendancy and now pervades condensed matter and nuclear physics as well as quantum chemistry. The Fourteenth RPMBT conference takes place within the centennial year of Eugene Feenberg’s birth. This gives us an excellent opportunity for remembrance of the man and celebration of his legacy for theoretical physics. The imprint of his intellect endures both in his foundational research in quantum many-body theory,

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beginning with his studies of nuclear forces and nuclei in the early days of nuclear physics and culminating in the Method of Correlated Basis Functions (CBF), and in the standards he set for microscopic (“ab-initio”) theory . There is also Feenberg the wise teacher, mentor, and exemplary role model, whose influence continues to be spread by his former students and colleagues, and by their own scientific progeny.

2. Birth and Youth Eugene Feenberg was born on October 6, 1906 in Fort Smith, Arkansas, to Polish immigrant parents. His father Louis, starting out as a peddler in his youth, traveled widely in the central states west of the Mississippi, settling for a while with his young wife Esther in Deadwood, South Dakota. He later achieved moderate financial success as a junk dealer in Fort Smith. At the time, this profession was not high on the social ladder, but — as Eugene himself has pointed out — its standing has risen with the increase of environmental concerns and the growing emphasis on recycling. Feenberg grew up in Fort Smith and attended the public high school, where he excelled in math and science, occupying himself with electrical gadgets, motors, and radios in his spare time. College was simply not part of his world, so after high school he worked for three years in a number of odd jobs (e.g. with his uncle’s fur business in Illinois). This rather dreary experience convinced him that he was not cut out for making a living in ordinary business jobs; he finally and firmly decided to pursue his true interests in science. His family had moved to Dallas, so he entered the University of Texas (UT) at Austin in 1926 (where the tuition was $25/semester), studying physics and math and making up for lost time by finishing first in his class with both B.S. and M.A. degrees in three years. He was the first graduate of his high school to attend college.

3. From Austin to Cambridge Feenberg’s brilliance attracted the attention of his professors at UT, including C. P. Boner and Arnold Romberg in Physics and Hyman Ettlinger and R. L. Moore in Math. With Boner’s support, he applied to Harvard, where he undertook doctoral studies during 1929-33. Early on, the Stock Market crash curtailed financial help from Eugene’s father, but Harvard physics faculty arranged part-time employment for him with Raytheon to fill the gap. Feenberg’s thesis research was directed by Edwin C. Kemble, who had established one of the early schools of theoretical physics in the U.S. The thesis, resulting in Feenberg’s first publication (1932), developed the quantum theory of scattering and contained the first statement and proof of the quantum optical theorem. During his years at Harvard, Eugene was also mentored by John Van Vleck, and he took courses from Kemble, John C. Slater, Percy Bridgman, and George Washington Peirce.

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4. European Hejira Harvard awarded Feenberg a Parker Traveling Fellowship in 1931, which allowed him to study in Europe for a year and a half. This was a time and place of turbulent developments both in physics and politics, as the ramifications of quantum mechanics were being explored and Nazism and Fascism emerged as horrific threats to world order and basic human rights. Eugene spent varying periods in Munich (Sommerfeld), Zurich (Pauli), Rome (Fermi), Berlin (Wigner), and Leipzig (Heisenberg). In hindsight, he remarked that he was not really mature enough to be sent over that way (on his own, without a definite study plan), but the experience at these magnificent sites of quantum ferment had to be inspiring. One highlight was the instant rapport he established with Ettore Majorana, a young Italian genius of Eugene’s age who disappeared inexplicably in 1938. By contrast, Feenberg reacted with shock and outrage to the political turmoil and violent anti-Semitism generated by the Nazi seizure of power in 1933. He could hardly restrain himself as he walked the crowded streets of Leipzig and saw roving gangs of Brown-Shirt thugs attack Jewish shops and their owners — passersby warned him to keep calm. Upon receiving a letter from Eugene describing this experience, Kemble immediately sent a plea for him to return to the U.S., fearing for his safety. 5. Precocious Nuclear Shells and the First Internal Symmetry After his return, Feenberg spent two years (1933-35) at Harvard as an instructor, in a holding pattern until a regular faculty appointment could be found for him. With the Great Depression in full effect, faculty jobs were scarce, and in addition there was a pernicious anti-Semitism within academic administrations. (Presumably to dispel the stereotype, Kemble is known to have described Feenberg as a “tall, rangy Texan.”) It was in this period that Eugene carried out the first calculations of the structure of the lightest nuclei (d, t, 3 He, α) with postulated nuclear forces — variational calculations based on trial functions built from Gaussians. Van Vleck came to him one day with the suggestion that such calculations should be carried out. Gene pulled open his desk drawer and handed Van Vleck the results. A one-year faculty slot at Wisconsin opened up for Feenberg in 1935-36 when Gregory Breit, already a prominent nuclear theorist, left for a visit to Princeton. At Wisconsin, Feenberg shared an office with Eugene Wigner, and there they began laying the foundation for the nuclear shell model with calculations on p-shell nuclei that led ultimately to supermultiplet theory, a precursor of supersymmetry. In 1936, experimental results for proton-proton scattering broke open the controversial puzzle of the like-nucleon interactions, supporting the charge independence of nuclear forces and giving the first solid evidence of an internal symmetry: isospin. Breit and Feenberg, through correspondence, published one landmark Physical Review paper on the subject, back-to-back with another by Cassen and Condon. Steven Weinberg has called attention to the historical importance of these papers.

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Wigner next arranged a two-year staff appointment for Feenberg at the Institute for Advanced Study (1936-38). There Eugene moved the p-shell calculations forward based on the new symmetry principles, working with Wigner and supervising the research of Melba Phillips. Phillips, one of Oppenheimer’s first students, was among the very few women theorists on the scene in the 1930’s; she was later to gain international prominence in science education. While at the Princeton Institute, Feenberg also collaborated with John Bardeen (then a Harvard Junior Fellow) on symmetry effects in nuclear level spacings. With the recommendations of Kemble, Wigner, and Isidor Rabi, in 1938 Feenberg was recruited to the faculty of New York University, where he began a long association with Henry Primakoff. And one day a vivacious young student named Hilda Rosenberg appeared at Gene’s office door, seeking support for a liberal political cause — and sparking a relationship that led to marriage. Two sons, Andrew and Daniel, were born to Hilda and Eugene; both have enjoyed successful intellectual careers, Andrew in philosophy and Daniel in economics. There are two grandchildren. Hilda Feenberg passed away in 1997.

6. Selective War Work During World War II, Feenberg took a leave of absence from NYU to join the Allied war effort, engaging in radar research at the Sperry Gyroscope Laboratories on Long Island. There he applied electromagnetic theory to microwaves and the development of Klystron tubes and is credited with an important technical innovation. Gene was invited to join the fission bomb project at Los Alamos but declined; this is fully consistent with his expressed views on the responsibility of scientists to make morally correct choices in applying their knowledge. It is worth noting here that, quite independently of Meitner, Eugene worked out and published (in a 1939 letter to Phys. Rev.) the standard analysis showing how the competition between Coulomb and surface energies governs the possibility of nuclear fission.

7. At Home in St. Louis with Mature Nuclear Shells When Arthur H. Compton — who directed the Metallurgical Project in the development of the fission weapon — returned to Washington University (WU) as Chancellor in 1945, he spearheaded a significant expansion of the science departments. Both Feenberg and Primakoff moved from NYU to join the physics faculty as associate professors in 1949. With the war years behind them, physicists resumed a vigorous engagement with fundamental scientific issues, armed with fresh ideas and technological advances in instrumentation. Nuclear facilities developed in connection with the war effort produced copious data that cried out for explanation in terms of nuclear models. Feenberg seized the opportunity to gain leadership in reviving the shell model as a viable alternative to the collective (or compoundnucleus) model of Bohr, which had dominated thinking after the discovery of fission and measurements of neutron cross sections at low energy.

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It is telling that in Blatt & Weisskopf’s monumental 1952 text Theoretical Nuclear Physics, it was Bethe, Breit, Feenberg, and Wigner who were the winners of the citation contest. Analyzing the new data on isomerism, beta decay, spins, and magnetic moments in the late 40’s, Feenberg provided compelling evidence for the validity of the shell model, summarized in his 1955 monograph Shell Theory of the Nucleus (written while on leave as Higgins Visiting Professor at Princeton (195354)). But he missed one vital point: the existence of a spin-orbit component in the nuclear potential well, which leads naturally to the observed magic numbers for shell closure. Both J. Hans D. Jensen and Maria Goeppert-Mayer got this part right (the idea was actually suggested to Goeppert-Mayer by Fermi). They shared half of the 1963 Nobel Prize in Physics for the nuclear shell model, with Wigner receiving the other half for his work on fundamental work on symmetries in quantum mechanics. In retrospect, the magnitude of Wigner’s multifaceted achievements warranted an unshared prize, and Feenberg, for his pioneering work both before and after WWII, deserved a share of the shell-model award. This circumstance may be the origin of Gene’s incisive observation, “Any physicist who misses a chance to be magnanimous is a fool.” This statement, which we should all take to heart, is a poignant reflection of Eugene Feenberg’s stoicism, dignity, and generosity.

8. A Man of Few Words — But Many Ideas Feenberg and Primakoff shared an office at Washington University, face-to-face with their desks pushed together in the middle of the room. This proximity led to some highly original and imaginative science. With Compton a Washington University “franchise player” (the original Compton experiment was performed on the WU campus in 1922), and with Arthur Holly physically present either as Chancellor — and later as Professor-at-Large — it was natural for Gene and Henry to propose and analyze the inverse Compton effect (protons and electrons scattering off photons). The result was a truly classic 1948 paper in The Physical Review. Inverse Compton scattering is now a staple of high-energy astrophysics. Another idea they put forward in 1946 (pre-Bodmer, very pre-Witten) is that nuclei are metastable and can collapse into abnormal, superdense matter. Modest and thoughtful, Eugene Feenberg was a quiet man of few but well-chosen words, which could be filled either with wisdom or humor (or both). Conversation with him could be somewhat halting, as he tended to ponder deeply and at length, especially when a scientific matter was under discussion. Accustomed to this trait, I took some malicious pleasure in watching the obvious discomfort of visitors, when Gene lapsed into silence extending to minutes, before speaking directly to the point. Many of us have had the experience of being interviewed by an FBI agent (or other government official) in the process of security clearance of a scientist known to us. It came to pass that such an agent interviewed Eugene to determine the loyalty and sobriety of Henry Primakoff. Eugene gave brief but reassuring answers to the lengthy series of questions, and, at the end, told the agent, “Dr. Primakoff is really

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a very fine man. Would you like to meet him? He’s sitting right across from us.” In 1950 Robert Hofstadter drove with his family from Princeton to Stanford, where he had accepted a faculty position. Passing through St. Louis, they visited Eugene and Hilda. During a extended account by Bob of his plans for experiments using his new NaI(Tl) crystals, Gene suggested, “Why not do electron diffraction [on nuclei] like the earlier work on atoms?” This question was the catalyst for Hofstadter’s research at Stanford leading to measurements of charge and magnetic moment distributions in nuclei and nucleons, recognized in a 1961 Nobel award.

9. Birth and Youth of CBF Theory From the early 50’s into the middle of that eventful decade, Feenberg’s interests were evolving from nuclear structure theory toward something recognizable as modern microscopic many-body theory. The pattern is evident in his elegant papers of this period analyzing different formulations of perturbation theory (with significant precursors in 1948). It was toward the end of this transitional period, in Fall 1956, that I entered the physics graduate program at Washington University, having been attracted by the opportunity of research in nuclear theory under Feenberg’s direction. At that time, Feenberg and Primakoff formed the department’s graduate admissions committee. Looking back, I realize that on seeing my application, coming as it did from the University of Texas (UT), Feenberg must have had a sense of déj` a vu. As Eugene had been, I was a student on a fast track to the B.S. and M.A., and although nearly thirty years had elapsed, I had been taught by some of same professors, most notably Ettlinger and Moore. (Also, the tuition was still $25/semester! The legendary mathematician R. L. Moore continued to teach for another 13 years in his famously crusty Socratic style, until he was forced to retire in 1969 at age 86+. Another curiosity: Feenberg was at UT at the same time my mother studied there, and they probably had the same professor — Romberg — for the general physics course.) Together, Feenberg and I studied the advances being made in microscopic theories of quantum many-body systems based on methods, diagrammatic and otherwise, borrowed from quantum field theory (Goldstone, Hugenholtz, Hubbard, Bloch & De Dominicis, Pines, the Russian School, ...). We also followed the development of Brueckner’s more rough-and-ready reaction-matrix theory, based on resumming ladder diagrams of perturbation theory including medium dispersion in the propagators. (Brueckner visited St. Louis for a week in 1957, giving a series of lectures on his work; the notes were made available in Brueckner’s chapter of the 1959 Les Houches Summer School volume on The Many-Body Problem.) However attractive these approaches might be, we chose another direction. In view of Feenberg’s earlier work, our approach was quite naturally based on a wave-function description in which the most important geometric correlations are included at the outset — arguably a superior strategy when dealing with interactions that feature a strong inner repulsive core. The first paper on what was to become the Method of Corre-

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Fig. 1. Eugene Feenberg (center) conversing with Joseph Hirschfelder (left) and Richard Norberg (right) in the Pfeiffer Physics Library during the 1974 Washington University symposium celebrating Feenberg’s career.

lated Basis Functions (CBF) was published in The Physical Review in 1959 — this was in fact the only paper we coauthored. 10. CBF Theory — Halcyon Days In 1964, Eugene Feenberg succeeded Edward Uhler Condon as the fifth Wayman Crow Professor of Physics. Dating from the 1860’s, this is the oldest endowed chair at Washington University, held previously by Compton. The last two decades of Feenberg’s professional life were devoted to the development and application of CBF as a practical and comprehensive scheme for quantitative description of strongly interacting quantum many-body systems. While the main focus of Eugene and his students was on the helium quantum fluids, advances were also made for Coulomb systems. In 1963 I returned to Washington University as a faculty member after postdoctoral study at Princeton and in Europe, and undertook the application of CBF to nuclear problems. The majority of Ph.D. students supervised by Feenberg during his long and productive career worked on projects in CBF theory. In another case of déj` a vu, two more Texans followed me from Austin to St. Louis to work with Feenberg: Tollie Davison and H. Woodrow (Woody) Jackson. (This repeated a pattern observed thirty-some years before, with three Texas students following Feenberg’s path from Austin to Harvard — Noyes Smith, Charles Fay, and Arnold Romberg’s son.) Indeed, quite a collection of talented students were to benefit from Feenberg’s guidance and

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care as research mentor, including Clayton Williams (from Tulsa; almost a Texan), Fa Yueh (Fred) Wu, William Mullin, Walter Massey, Chia-Wei Woo, Deok Kyo Lee, Hing-Tat Tan, Charles Campbell, David Hall, and Kai-Yaun Chung. (Precursors in the transitional period were Mark Bolsterli and Paul Goldhammer.) After these there are flocks of scientific grandchildren and great-grandchildren, too numerous to list. If postdoc mentoring links are counted, we can include Manfred Ristig, Eckhard Krotscheck, Stefano Fantoni, Klaus Gernoth, and Arturo Polls, among other active figures on the many-body scene. We can all be proud of our heritage, while continuing to honor and preserve Feenberg’s legacy in our work. Feenberg retired in 1975 at the age mandated at that time. In the same year, he was elected to the U.S. National Academy of Sciences. On November 7, 1977, he died of a heart attack suffered while walking home from his office. In his lifetime, he had been to Europe only three times. He was looking forward to attending the First International Workshop on Recent Progress in Many-Body Theories, scheduled for Trieste in 1978. The meeting was dedicated to his memory, and the Feenberg Medal for Many-Body Physics was established at RPMBT3 in 1983. 11. Conclusion If Eugene were able to read the above narrative of his life course, he would be embarrassed by the praise, but show his appreciation by variously telling us to “Tend to your knitting!,” and, in Texan vernacular, “Come back full of beans!” I close with a short message to the 2007 Feenberg Medalists, Stefano Fantoni and Eckhard Krotscheck: At the Centenary of Feenberg’s birth, no two theorists could better represent his legacy and realize his aspirations. Acknowledgments and Bibliographical Notes I am grateful to the Niels Bohr Library of the American Physical Society’s Center for the History Physics, for making available an electronic copy of the Eugene Feenberg Oral History Interview of April 13, 1973, carried out by Charles Weiner. Other valuable resources include the entries for Feenberg and for Robert Hofstadter prepared for the Biographical Memoirs of the National Academy of Sciences by the late George Pake and by J. I. Friedman and W. A. Little (www.nap.edu/readingroom/books/biomems/efeenberg and /rhofstadter), respectively. The “tall, rangy Texan” quote is from Daniel J. Kevles’ The Physicists: The History of a Scientific Community in Modern America (A. A. Knopf, New York, 1978). The obituary by K. A. Brueckner, J. W. Clark, and H. Primakoff in Nuclear Physics A 328, 1 (1979) includes an (almost) complete listing of Feenberg’s publications.

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STEFANO FANTONI: FEENBERG MEDALIST 2007 MICROSCOPIC MANY-BODY THEORY OF STRONGLY CORRELATED SYSTEMS A. POLLS Departament d’Estructura i Constituents de la Mat` eria, Universitat de Barcelona, Avda. Diagonal 647, E-08028 Barcelona, Spain ∗ E-mail: [email protected] The Eleventh Eugene Feenberg Medal is awarded to Stefano Fantoni in recognition of his leading role in the development and extensive application of correlated wave function approaches, including the advance of Fermi hypernetted chain theory, thereby providing an accurate, quantitative, microscopic description of strongly interacting quantum manyparticle systems, especially for nuclear systems.

Stefano Fantoni was born in 1945 in Taranto (Italy). He received his PhD degree in 1971 from the Scuola Normale Superiore in Pisa under the supervision of Sergio Rosati. There followed a fruitful period (1971-1986) in which he held an appointment as Associate Professor of Physics in Pisa, but also spent several long research stays abroad. These included visits to the Neils Bohr Institute and the University of K¨ oln. He spent two years (1981-82) as Visiting Associate Professor at the University of Illinois-Urbana, establishing ties with the Urbana many-body group, leaded by Prof. Vijay Pandharipande. In 1986 he was named to a full professorship at the University of Lecce. Making an important career move to SISSA in Trieste in 1992, and he began to combine research with duties in scientific management, a mode of operation well suited to his quick mind, boundless energy, and consummate organizational skills. SISSA proved to be an ideal environment for his dual role. Fantoni has served as Director of the Interdisciplinary Laboratory (1992-2000) and of the SISSA Master in Science Communication (1994-2000), becoming director of SISSA in 2004. It is testimony to both the scope of his abilities and the strength of his dedication that these administrative activities have had negligible impact on his research creativity and output. The derivation of FHNC equations for resummations of cluster diagrams, achieved independently by Krotscheck&Ristig and Fantoni&Rosati in 1974-75, 1 was a breakthrough that enabled many-body theorists to perform realistic and accurate ab-initio calculations for strongly interacting quantum systems in both con-

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Stefano Fantoni

densed matter and nuclear physics. Fantoni has made profound contributions to the methodological aspects of correlated wave functions theories. Stefano’s colleagues continue to marvel at his unique ability to translate into diagrammatic language the expansions of quantities that are otherwise intractable. “Give him a physical quantity to calculate and he brings back a diagrammatic expansion” — which he has re-summed into calculable form. Fantoni has shown us how to fulfill the promise of the CBF approach as both a practical and accurate tool of ab-initio many-body theory, comprehensive in its scope. In this respect, Fantoni’s contributions are destined to have lasting impact across a broad range of subfields of physics. As he likes to say : Nearly all physics is many-body physics at the most microscopic level of understanding . Once the FHNC scheme had become routine for the ground states (at least for simple systems), Fantoni turned to the evaluation of more sophisticated quantities including the self-energy of nucleons. This major extension of the CBF required the acquisition of new insights into the perturbation theory within a basis of correlated wave functions. In particular, he calculated the imaginary part of the nucleon self-energy due to the coupling to two-particle, one-hole and one-particle, two-hole excitations.2 This computational feat was performed during Stefano’s intense collaboration with Pandharipande, with whom he also analyzed the momentum distribution of nuclear matter,3 providing a quantitative estimate of the depletion below the Fermi momentum. The same period saw a remarkably successful application of the CBF approach to the single-particle spectrum of a 3 He impurity in liquid 4 He obtaining a 3 He effective mass in agreement with experimental data.4

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Based on these formal developments, Fantoni, united to Omar Benhar and Adelchi Fabrocini into a very strong team, has spearheaded the application of CBF methods to microscopic description of the structure and dynamics of nuclear matter and finite nuclei, as well as neutron-star matter. The determination of singleparticle Green’s functions and the longitudinal response of nuclear matter within CBF framework have played an extremely important role in theoretical interpretation of spectroscopic factors measured in (e,e’p) experiments and generally the measured response of nuclei to electromagnetic probes.5,6 Stefano also takes great pleasure in constructing HNC or FHNC equations for different kinds of correlated wave functions, deriving closed sets of integral equations for the calculation of relevant observables. In this spirit, he applied the CBF approach to fermions in a lattice7 and also formulated the FHNC theories for the Gutzviller-correlated wave functions8 and more recently we can notice his studies on Laughlin quantum Hall states using HNC techniques.9 Notwithstanding, the impressive developments in microscopic many-body theory represented by FHNC and advanced CBF technology, there remain formidable obstacles to quantitative treatment of problems in which elementary diagrams contribute strongly or the non-commutativity of state-dependent (e.g., spin-dependent) correlations becomes a serious issue. Working mainly with Kevin Schmidt, Fantoni has taken important steps toward avoiding these difficulties by means of a new Monte Carlo method. Known as the Auxiliary Field Diffusion Monte Carlo (AFDMC) algorithm,10 this method is well suited to the treatment of Hamiltonians containing tensor and spin interactions as arise in nuclear physics. The AFDMC method has shown considerable promise in several applications in nuclear physics.11,12 Also deserving special mention is the comprehensive study of pairing in nuclear systems, which compares results for superfluid gaps obtained from the AFDMC method and from CBF theory.13,14 Finally, we should not neglect a set of important contributions to the field of Cold Atoms made by Fantoni in collaboration with A. Smerzi, even though this work has no direct relation to FHNC or CBF. He has made his mark in masterful studies of the Josephson effect in two weakly linked Bose–Einstein condensates in the framework of the time-dependent Gross–Pitaevskii equation.14 As already suggested by his administrative roles at SISSA, Stefano Fantoni has long maintained an intense —even frenetic— engagement in the promotion of science and in the political responsibility of scientists. ”Popularization of science is very important because increases the value of democracy ” and ”It is a duty of any scientist to participate at the political level because he is a citizen and in addition he is a very privileged citizen due to his culture” are sentences that pronounced with the credibility of Stefano become a source of constant motivation for social compromise. He was a founder of the Elba International Physics Center (EIPC) at isola d’Elba in Italy. He was also one of the founders of the European Centre for Theoretical Studies in Nuclear Physics and Related Areas (ECT*) in Trento, which

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has been of crucial importance in the maintenance and enhancement of theoretical nuclear physics, both in Europe and worldwide. Fantoni has participated also in the Advisory Program Committees (PACs) of several facilities, including Jefferson Laboratory (CEBAF) and the Legnaro Laboratories of INFN. Within the Many-body community, he has been active on the standing Advisory Committee of Conferences on Recent Progress in Many-Body Theories, and is a member of the Editorial Board of the World Scientific series on Advances in Quantum Many-Body Theory. In line with his commitment with the popularization of science, he was the founder of the first Italian Master in Science Communication in SISSA. He was also serving as a member of the Governmental Committee on “Public Understanding of Science” (1995-1997) and Vice-President of the “Fondazione sulla libert´ a delle Scienze ”. For his contributions to popularization of science he got the 2001 Kalinga prize, awarded by UNESCO. Stefano Fantoni has attracted —and often mentored— many talented collaborators who helped him to built impressive bodies of work. Under Fantoni’s leadership, SISSA became a magnet for both students and mature scientists eager to contribute to many-body physics. Thus other coworkers that should be named include E. Tosati, I Sick, S. Moroni, F. Pederiva, L. Reatto, O. Ciftja, S. Gandolfi, A. Sarsa, Ll. Brualla, and R. Guardiola. Finally, and looking in to the future I want to say that Stefano’s creativity and motivation ensure many new contributions to science.

References 1. S. Fantoni and S. Rosati, The Hyper–Netted-Chain approximation for fermion systems, Nuovo Cimento, A25, 593 (1975). 2. S. Fantoni, B.L. Friman, and V. R. Pandharipande, Microscopic calculation of the imaginary part of the nucleon optical potential, Nucl. Phys. A386, 1 (1982). 3. S. Fantoni and V. R. Pandharipande, Momentum distribution of nucleons in nuclear matter, Nucl. Phys. A427, 473 (1984). 4. A. Fabrocini, S. Fantoni, S. Rosati, and A. Polls, Microscopic calculation of the excitation spectrum of one 3 He impurity in liquid 4 He, Phys. Rev. B 33, 6057 (1986). 5. O. Benhar, A. Fabrocini and S. Fantoni, The nuclear spectral function in infinite nuclear matter, Nucl. Phys. A505, 267 (1989). 6. O. Benhar, A. Fabrocini, S. Fantoni, and I. Sick, Scattering of few GeV electrons by nuclear matter, Phys. Rev. C 44, 2328 (1991). 7. X.Q. Wang, S. Fantoni, E. Tosatti, L. Yu, and M. Viviani, Correlated Basis Function method for fermions on a lattice- The one-dimensional Hubbard model , Phys. Rev. B 41, 11479 (1990). 8. X.Q. Wang, S. Fantoni, E. Tosatti, L. Yu, Fermi-Hypernetted-Chain scheme for Gutzwiller correlated wave functions, Phys. Rev. B 49, 10027 (1994). 9. O. Ciftja, S. Fantoni, Application of Fermi-hypernetted-chain theory to compositefermion quantum Hall states, Phys. Rev. B 56, 13290 (1997). 10. KE Schmidt and S. Fantoni, A Quantum Monte Carlo method for nucleon systems, Phys. Lett. B 446, 99 (1999). 11. S. Fantoni, A. Sarsa and KE Schmidt, Spin susceptibility of neutron matter at zero temperature, Phys. Rev. Lett, 87, 81101 (2001).

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12. S. Gandolfi, F. Pederiva, S. Fantoni and K. E. Schmidt, Quantum Monte Carlo Calculations of Symmetrical Nuclear Matter, Phys. Rev. Lett. 98, 102503 (2007). 13. S. Fantoni, Correlated BCS theory, Nucl. Phys. A363, 381 (1981). 14. A. Fabrocini, S. Fantoni, A. Yu. Illarionov, K.E. Schmidt, 1 S0 Supefluid Phase Transition in Neutron Matter with Realistic Nuclear Potentials and Modern Many-Body Theories, Phys. Rev. Lett. 95, 192501 (2005). 15. A. Smerzi, S. Fantoni, S. Giovannazzi, S.R. Senoy, Quantum Coherent Tunneling between two trapped BEC, Phys. Rev. Lett. 79, 4950 (1997).

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ECKHARD KROTSCHECK: FEENBERG MEDALIST 2007 MICROSCOPIC MANY-BODY THEORY OF QUANTUM FLUIDS M. SAARELA Physical Sciences, University of Oulu, P.O.Box 3000, FIN-90014 University of Oulu, Finland ∗ E-mail: [email protected] The Eleventh Eugene Feenberg Medal is awarded to Eckhard Krotscheck in recognition of his leading role in the development and extensive applications of the correlated basis function method, including the advance of Fermi hypernetted chain theory, thereby providing an accurate, quantitative, microscopic description of strongly-interacting quantum many-body systems, especially for inhomogeneous quantum fluids.

Eckhard Krotscheck was born in 1944 in Germany, but soon his family moved to Southern Austria where Eckhard spent his childhood, learned to ski and enjoy mountains. Upon returning to Germany with his family at the age of eight, Eckhard started his climb through German schools. He received the Diplom in theoretical physics from the University of Cologne in 1971 under the supervision of Professor Peter Mittelstaedt and continued his studies in Cologne in quantum many-body theory with Professor Manfred Ristig up to the doctoral degree in 1974.1 Particularly noteworthy of Eckhard’s research during this period was the development of the Fermi hypernetted chain method, a major advance in the theory of strongly correlated fermion quantum fluids. His first post doctoral position was an assistantship in the Institute for Theoretical Physics in Hamburg 1975-1981, where he continued his work with the variational many-body theory and made very important contributions to the method of correlated basis functions.2 During this period he also worked with Wolfgang Kundt on problems in relativistic astrophysics. These activities lead to the Habilitation in 1979. One year later he was appointed to a Heisenberg fellowship for six years, which opened the possibility of more extended international collaborations. The first contact with Gerry Brown came on a visit to Nordita in 1978, which led to an invitation to Stony Brook for two years (1979-81) and to a long-standing, fruitful collaboration with Andy Jackson and Roger Smith. Eckhard’s career continued in the U.S.A. at the University of Illinois at Urbana-Champaign (1981-82) with David Pines and in Santa Barbara (1982-83) with Walter Kohn, both former recipients of the Feenberg medal. For one year Eckhard returned to Germany to the Max Planck Institute in Heidelberg (1983-1984) before moving to the faculty at Texas A&M in 1984, where he became a full professor of physics in 1988. During

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Eckhard Krotscheck

those years he developed very close contacts with Chuck Campbell in Minneapolis and John Clark in St Louis, with numerous, mutual visits linking him to the extraordinary school of theorists initiated by Eugene Feenberg. In 1995 Eckhard returned to Austria as he was appointed to the Professorship in Theoretical Physics at the Johannes Kepler University of Linz, where he is presently Chair in Theoretical Physics. Eckhard Krotscheck solved several long-standing, fundamental problems in the microscopic theory of inhomogeneous quantum fluids and fluid mixtures. His seminal work on inhomogeneous quantum Bose fluids in collaboration with Kohn and Qian3 turned the unsolved variational many-body problem of a Boson film adsorbed on a substrate into a numerically tractable form. He and his collaborators published a critically important series of papers that provided the basis for a general theory of inhomogeneous quantum-many body systems. The key idea was to expand the complicated integral equation optimizing the correlation functions in terms of the Feynman phonon basis, which made explicit numerical calculations possible. Since then the field has flourished, expanding into detailed treatment of the growth mechanisms and dynamics of thin films, including impurities, scattering, and temperature dependence,4,5 as well as the dynamics of helium droplets and confined 4 He with impurities.6–8 The quantitative calculations performed on these complicated systems, enhanced with perceptive analysis of the results, have given crucial feedback and guidance to leading experimentalists. The key ingredients of a successful microscopic variational many-body theory are (i) the quality of the wave function, (ii) an astute identification of the sets of

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important diagrams, and (iii) an accurate solution of the Euler equations that optimizes the correlation functions. In numerous practical implementations, Eckhard Krotscheck has been an undisputed master of all three of these aspects. This mastery has led, in particular, to very accurate microscopic descriptions of 3 He-4 He mixtures9 and electron and charged-boson gases. A wide variety of experimental findings on magnetic susceptibility, effective mass, layered modes, etc., have received detailed explanations based on the microscopic theory at its very best. 10 More recently he has directed intensive efforts in implementing ingenious, new algorithms for the density functional theory, which substantially reduce the simulation time,11,12 and in developing wave function based Euler–Lagrange theories in terms of the pair density functional theory as we have learned during this conference. Eckhard’s deep understanding of many-body systems ranging from condensed matter to nuclear physics, coupled with his passion and honesty in solving the finest numerical details has acquired him the leading role among many-body theorists. In addition to his prolific scientific work, Eckhard has also served the community of physicists through numerous conferences and meetings he has organized. His enthusiasm and effectiveness have always made it both exciting and pleasing to take part in these encounters.

References 1. E. Krotscheck and M. L. Ristig, Hypernetted Chain Approximation for Dense Fermi Fluids, Phys. Lett. 48A, 17 (1974); E. Krotscheck and M. L. Ristig, Long Ranged Jastrow Correlations, Nucl. Phys. A 242, 389 (1975). 2. E. Krotscheck and J. W. Clark; Studies in the Method of Correlated Basis Functions: III. Pair Condensation in Strongly Interacting Fermi Systems, Nucl. Phys. A333, 77 (1980). 3. E. Krotscheck, G.-X. Qian and W. Kohn, Theory of inhomogeneous quantum systems I: Static properties of Bose fluids, Phys. Rev. B 31, 4245 (1985). 4. B. E. Clements, E. Krotscheck and M. Saarela, Impurity dynamics in boson quantum films, Phys. Rev. B 55, 5959 (1997). 5. C. E. Campbell, E. Krotscheck and M. Saarela, Quantum sticking, scattering, and transmission of 4 He Atoms from superfluid 4 He surfaces, Phys. Rev. Lett. 80, 2169 (1998). 6. S. A. Chin and E. Krotscheck, Systematics of pure and doped 4 He clusters, Phys. Rev. B 52, 10405 (1995). 7. E. Krotscheck and R. E. Zillich, The Dynamics of 4 He Clusters, J. Chem. Phys. 115, 10161 (2001). 8. V. Apaja and E. Krotscheck, Layer- and bulk-roton excitations of 4 He in porous media, Phys. Rev. Lett. 91, 225302 (2003). 9. E. Krotscheck and M. Saarela, Theory of 3 He-4 He mixtures: Energetics, structure and stability, Phys. Rep. 232, 1-86 (1993). 10. E. Krotscheck, M. Saarela, K. Sch¨ orkhuber, and R. Zillich, Concentration Dependence of the Effective Mass of 3 He Atoms in 3 He-4 He Mixtures, Phys. Rev. Lett. 80, 4709 (1998). 11. M. Aichinger and E. Krotscheck, A fast configuration space method for solving local Kohn–Sham equations, Computational Materials Sciences 34, 188 (2005).

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12. M. Aichinger, S. A. Chin and E. Krotscheck, Fourth–order algorithms for solving local Schr¨ odinger equations in a strong magnetic field, Computer Physics Communications 71, 197 (2005).

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FRANK VERSTRAETE: HERMANN KUEMMEL AWARD 2007 G. ORTIZ Department of Physics, Indiana University, Bloomington, IN 47405, USA ∗ [email protected]

It is a great pleasure and undeserved honor to be in front of you presenting Frank Verstraete as he is awarded the Hermann Kuemmel early achievement award in Many-body Physics. The pleasure is multiple. On one hand, Frank is its first recipient and on the other Professor Hermann Kuemmel is present among us. As I will argue in the following, Frank has set a standard of excellence that it will be very difficult to match in future awards. Moreover, I believe and hope that he will become a role model for generations of young physicists who envision unconventional approaches to attack Physics problems but are afraid to pursue them because of conventional wisdom. Needless to mention that Science needs provocative and unbounded young minds able to unknot the bundle of surprises Nature confront us with. Frank was born in Izegem, Belgium, in November 1972. He did undergraduate studies in Electrical Engineering in Leuven and two years after graduation he completed studies in Theoretical Physics at the University of Ghent. Having decided that he better enjoyed unraveling the mysteries of Nature rather than moving or wiring electrons he started doctoral studies on quantum information theory back in Leuven, under the supervision of Professors De Moor and Verschelde. He completed his PhD dissertation in October 2002 with important results on optimal teleportation protocols in the presence of noise. Then he saw the light! First he moved to the Max Planck Institute for Quantum Optics in Garching, and then he realized that the strongly interacting quantum many-body problem is an interesting problem worth exploring. While in Germany he did seminal work on localized entanglement and its connection to decaying correlations in quantum phase transitions and also work on the relation between entanglement and the complexity of simulating strongly interacting systems together with Ignacio Cirac and several coworkers. I met Frank in 2004, while visiting Ignacio Cirac to discuss problems of common interest in quantum critical phenomena. I still remember the two and something hour seminar I gave where the big boss and his right hand, Frank, were constantly interrupting and asking me all kind of interesting questions about quantum correlations. One question that struck me as particularly interesting and challenging was: How do you characterize correlations

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Frank Verstraete

in a topologically quantum ordered state? Of course, I did not have the answer at that time and I must confess still don’t but I can guarantee you that the ghost of Frank still haunts me. In October of 2004, he decided to join the Institute for Quantum Information at CALTECH headed by John Preskill. For those of you not familiar with the sociology in the field CALTECH is a sort of Mecca for quantum information and John Preskill is the guru that, for example, made Stephen Hawking concede in public that Hawking was wrong regarding the loss of information in black hole physics, information loss does not occur after all. The time came to look for a stable job in Physics and I still remember trying to naively attract Frank to come and join the quantum information effort at Los Alamos. The time was the early months of the year 2006 and one could see Frank’s metamorphosis from a gifted unnoticed young man to the David Beckham of Physics: Every year, in the USA, there is a young star that gets most of the tenure-track faculty position offers and in the year 2006 Frank was the Chosen One! He got offers from all over the place including, MIT, Illinois (Urbana-Champaign), University of California, and many others. These circumstances helped him to immediately get a Full Professor position at the University of Vienna, a place that hosted unremarkable people such as Boltzmann and Schr¨ odinger, and which is currently a key center for quantum physics. Let me try to put Frank’s research work in context. Frank has been trained in the field of Quantum Information and Computation. This field studies how various tasks in computation and communication can be accomplished using a quantum

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physics representation of information. The key idea is to exploit and take full advantage of the fundamental laws of Nature with the hope that this is the best one can do. In 1935 Einstein, Podolsky, and Rosen identified the property of entanglement as the key feature of quantum phenomena which has no obvious classical analogue. Today we know about the importance of these non-local correlations as a defining resource for secure quantum communications, and moreover we believe that entanglement is an essential ingredient to understand and unlock the power of quantum computation. Frank’s research work in connection to the award centers on developing and using techniques borrowed from quantum information theory to classically simulate strongly interacting quantum systems. In particular, let me briefly mention what I consider to be the main insight Frank brought to the field at the risk of oversimplifying the magnitude of his contributions. There was a bottleneck in renormalization group methods such as DMRG in simulating systems in space dimensions larger than one. Frank realized that the fixed point physics captured by Steve White’s approach could never be efficiently extended to higher dimensions. Insights coming from his deep understanding of the way Matrix Product States work led him to propose new classes of states (PEPS) that more efficiently capture the fixed point physics. In other words, he made the DMRG method a computationally viable method in high space dimensions. Obviously, this is only a small selected sample of his outstanding work. Despite his short life in Science he has a very impressive publication record with approximately 50 published papers in first rate peer-reviewed journals including 22 Physical Review Letters and more than 1000 citations. Thinking outside the box is a phrase that is used to mean looking at a problem from a fresh and non-standard perspective. There are clearly many examples in human history and, in particular, in Physics of such lateral thoughts. In connection to Frank, I would simply like to quote Anton Zeilinger’s remark: A strong asset of Frank Verstraete is his ability to look at Physics problems from different angles. It is indeed this ability that allowed him to make such superb contributions. I would like to conclude with the hope that this award will add luster to his already productive career and, most importantly, will strongly motivate Frank to expand the great scientist he already is. Hermann you can rest assured that the prize is in very good hands.

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QUANTUM MONTE CARLO CALCULATIONS FOR NUCLEI AND NUCLEAR MATTER S. FANTONI1,2,∗ , S. GANDOLFI3,4 , F. PEDERIVA3,2 , K. E. SCHMIDT5 1 International School for Advanced Studies, I-34014 Trieste, Italy INFM DEMOCRITOS National Simulation Center, I-34014 Trieste, Italy 3 Dipartimento di Fisica, Universit` a di Trento, I-38050 Povo, Italy 4 INFN, Gruppo collegato di Trento, Universit` a di Trento, I-38050 Povo, Italy 5 Department of Physics, Arizona State University, Tempe, AZ 85287 USA ∗ E-mail: [email protected] 2

We report on the most recent applications of the Auxiliary Field Diffusion Monte Carlo (AFDMC) method from light nuclei to nuclear matter. Recent calculations of the ground state energy of 4 He, 8 He, 16 O, 40 Ca and symmetric nuclear matter using the semi-realist two-body interaction, Argonne v60 , which includes tensor and tensor-τ forces, are presented and discussed. Comparison of the light nuclei results to those of Green’s function Monte Carlo calculations shows the high level of accuracy of AFDMC for both open and closed shell nuclei, particularly when used in conjunction with the fixed phase constraint (FP–AFDMC). The application to heavier nuclei and to nuclear matter demonstrates the FP–AFDMC uniqueness, amongst the Quantum Monte Carlo methods, in dealing with large nucleonic systems interacting via realistic nuclear potentials and with unprecedented accuracy. Discrepancies have been found with previous Fermi Hyper Netted Chain and Brueckner–Hartree–Fock calculations. Most interestingly, the nuclear matter calculations strongly indicate that many-body forces are very important even at experimental equilibrium density. Preliminary results for pure neutron matter in both normal and BCS phase with Argonne v80 plus Urbana IX three-nucleon interaction are also presented. Keywords: Quantum Monte Carlo; nuclear and neutron matter; medium heavy nuclei.

1. Introduction The improved accuracy of experimental data on nuclei, together with a rediscovered role of nuclear matter properties in the understanding of nuclear structure1–4 and several phenomena of astrophysical interest5–8 asks for deeper and deeper investigation of the nuclear many-body problem. It is widely recognized that nuclei and nuclear matter belong to the class of strongly correlated Fermi systems. As a consequence, many-body methods capable of incorporating the main features of NN correlations in their core, rather than treating them perturbatively, are required for quantitative studies. Fermi Hyper Netted Chain (FHNC) calculations based on Jastrow correlated wave functions,9–12 and, later on, improved upon by Correlated Basis Function (CBF) perturbative corrections,13–18 have unambiguously shown the importance of NN correlations in several static and dynamic properties of nuclei and

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nuclear matter. For instance, the quenching of single particle amplitudes for states below the Fermi surface and the corresponding spreading of the strength up to very high excitation energies have been assessed only after FHNC/CBF calculations of the optical potential,19 the momentum distribution,20,21 the Green’s Function22–24 and the longitudinal response25 of nuclear matter have been performed. Subsequent high accuracy electron scattering experiments at intermediate energies performed on various nuclei have confirmed the findings of such calculations. We are now facing, on one side, with the problem of finding a fundamental scheme for the description of nuclear forces, valid from the deuteron up to dense nuclear matter, and, on the other, with that of solving a many-body problem which is made extremely complex by the strong spin-isospin dependence of such forces. We know, from the work of Pandharipande and his collaborators, that threebody forces are necessary and almost sufficient to describe ground and low energy excited states of light nuclei (A ≤ 12).26–28 However, we do not know the role of m-body forces with m > 3 at increasing densities of nuclear matter. This is a fundamental problem, particularly in nuclear astrophysics, if we pretend, as we should, to be as model independent as possible in our theoretical analyses and predictions. Recently there have been a few attempts to reduce the problem of nuclear forces to a more fundamental level by exploiting in full the scheme of Effective Field Theory (EFT). However, this approach, at present, is applicable only to very small nuclei. 29 Integration of subnuclear degrees of freedom in EFT seems to provide interactions which are almost as accurate in describing NN scattering data as the most popular realistic interactions30 obtained in a semi-phenomenological way. We know that Quantum Monte Carlo (QMC) methods provide estimates of physical observables at the best known accuracy,31,32 and they are therefore very useful to gauge the validity of proposed interaction models without having the bias of using approximate methods. However, the strong spin-isospin dependence of the nuclear force seems to limit the use of standard QMC methods beyond A = 12.33 That is because QMC methods have to deal with the exponential increase in the computational time with particle number. For instance, in GFMC the spatial degrees of freedom are sampled, whereas the spin isospin degrees of freedom of the nucleons are explicitly summed up and not sampled. The exponential growth of the spatial degrees of freedom is controlled, but since there are four spin-isospin states per nucleon, the computations grow exponentially — roughly as four raised to the number of nucleons. This is a second very strong limitation in the theory of a modern theoretical nuclear physics, which aims to interface with astrophysics and particle physics. In this contribution we will not face the problem of nuclear forces, other than providing indications for the need of many-body forces, even at the equilibrium density of nuclear matter, but with that of performing accurate quantum simulations for large nuclei and nuclear matter. To make the QMC method computationally efficient, the spin-isospin degrees of

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freedom of the nuclear force must also be sampled. That is what the Auxiliary Field Diffusion Monte Carlo (AFDMC)34 does in the most efficient way known today. In this contribution we will show that AFDMC can be used to solve for the energy of nuclei with nucleons interacting via a semi-realistic two-body interaction which, however, does contain the tensor interaction and all the terms which make standard QMC approaches impractical. Adding the neglected spin-orbit terms and the threebody potential is not expected to change our main conclusion, namely that AFDMC is applicable with the same accuracy to both nuclei and nuclear matter. It has to be noticed that this achievement, if it will be proved to be true as we believe, will open up the possibility of increasing the accuracy of theoretical estimates on nuclei and nuclear matter at zero temperature of at least an order of magnitude. The original version of the AFDMC method34 included a path constraint35 (CP– AFDMC) to control the fermion sign problem. Such a method has been found to give reasonably good results for pure neutron matter (PNM both in the normal36–38 and in the BCS superfluid39 phases, neutron-drops40 and valence neutrons of neutronrich nuclei.41 However, when np and pp interactions are active, the strong tensor force in the isospin-singlet channel makes sampling the spin-isospin states more difficult, leading to unsatisfactory results.42 In this paper we demonstrate that a new version of AFDMC, based on the fixed phase approximation43,44 (FP–AFDMC), rather than on the path constraint, seems to overcome the tensor–τ problem, and, as a consequence, can be successfully applied to calculate the binding energy of large nuclei and nuclear matter with realistic potentials. Most of the results shown have been obtained by solving the following Hamiltonian H=

A A X X p2i + v60 (i, j) , 2m j>i=1 i=1

(1)

−1 where m−1 = (m−1 p + mn )/2, with mp and mn being the proton and neutron masses, and the two-body potential v60 (i, j) is a simplified version of the Argonne v18 interaction45 given by

v18 (i, j) =

A X

MX =18

vp (rij )O(p) (i, j) ,

(2)

j>i=1 p=1

where O(p) (i, j) are spin–isospin dependent operators. The Argonne v60 (i, j)46 twobody is obtained by projecting the Argonne v18 to the M = 6 level, so to describe the binding energy of deuteron. The six O (p) (i, j) terms in v60 (i, j) are given by the four central components 1, ~τi · ~τj , ~σi · ~σj , (~σi · ~σj )(~τi · ~τj ), the tensor Sij , and the tensor–τ component Sij ~τ · ~τj , where Sij = 3~σi · rîj ~σj · rîj − ~σi · ~σj . The inclusion of the other components, like the neutron-proton mass difference, the electromagnetic interactions and the spin-orbit interactions, as well as the three-body potential can be done with an increase in complexity. The energies of the alpha particle and the open shell nucleus 8 He have been calculated to test the accuracy of the fixed

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phase AFDMC by comparing them with those from GFMC.46 The same algorithm is applied to calculate the binding energy of 16 O, 40 Ca and symmetric nuclear matter, comparing with the available results obtained with Cluster Variational Monte Carlo,47 Brueckner–Hartree–Fock (BHF) in the two hole lines approximation and FHNC in the Single Operator Chain approximation (FHNC/SOC).48,49 The scheme of the paper is the following. A brief summary of the FP–AFDMC method is given in Section II. Section III and IV report and discuss the FP–AFDMC calculations for nuclei and nuclear matter respectively. The last section is devoted to conclusions and perspectives. 2. The FP–AFDMC Method Ground state AFDMC simulations rely, as do other traditional QMC methods, on previous variational calculations, often performed within FHNC theory, to compute a trial wave function ΨT , which is used to guide the sampling of the random walk. A typical form for ΨT is given by a correlation operator Fˆ operating on a mean field wave function Φ(R), hR, S|ΨT i = Fˆ Φ(R) .

(3)

Mean field wave functions Φ(R) that have been used are: (i) a Slater determinant ΦF G of plane wave orbitals for nuclear matter in the normal phase, (ii) a linear combination Φsp of a small number of antisymmetric products of single particle orbitals φj (~ri , si ) for nuclei and neutron drops, and (iii) a pfaffian Φpf , namely an antisymmetric product of independent pairs for neutron matter in superfluid phase. A realistic correlation operator is the one provided by FHNC/SOC theory, Q P (p) namely S j>i M (rij )O(p) (i, j), where S is the symmetrizer and the opp=1 f (p) erators O (i, j) are the same as those appearing in the two-body potential. Unfortunately, the evaluation of this wave function requires exponentially increasing computational time with the number of particles. This procedure is followed in variational and Green’s function Monte Carlo calculations, where the full sum over spin and isospin degrees of freedom is carried out. Since for large numbers of particles one cannot evaluate these trial functions, the much simpler correlation Q operator j>i f c (rij ), which contains the central Jastrow correlation only, is used instead. The evaluation of the corresponding trial function requires order A3 operations to evaluate the Slater determinants and A2 operations for the central Jastrow. Since many important correlations are neglected in these simplified functions, we use the Hamiltonian itself to define the spin sampling. The AFDMC method works much like Diffusion Monte Carlo.31,34,36,40,41 The wave function is defined by a set of what we call walkers. Each walker is a set of the 3A coordinates of the particles plus a number A of four component spinors each representing a spin-isospin state. The imaginary time propagator for the kinetic energy and the spin-independent part of the potential is identical to that used

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in standard diffusion Monte Carlo. The new positions are sampled from a drifted Gaussian with a weight factor for branching given by the local energy of these components. Since they do not change the spin state, the spinors will be unchanged by these parts of propagator. To sample the spinors we first use a Hubbard–Stratonovich transformation to write the propagator as an integral over auxiliary fields of a separated product of single particle spin-isospin operators. We then sample the auxiliary field value, and the resulting sample independently changes each spinor for each particle in the sample, giving a new sampled walker. Specifically, we write a v6 -type interaction as a sum of its scalar component v1 plus the spin-isospin dependent part Vsd , which includes the other five components v2−6 and is conveniently expressed in terms of three matrices 1X 1 X (σ) (στ ) (τ ) Vsd = σiα Aiα,jβ σjβ + σiα Aiα,jβ σjβ ~τi · ~τj + A ~τi · ~τj (4) 2 2 i,j i,j iα,jβ

where roman indices stand for nucleons while Greek indices indicate Cartesian components. The A matrices depend only on the positions of the particles. They are zero when i = j and they are real and symmetric so that they have real eigenval(σ) (στ ) (τ ) (σ) (στ ) (τ ) ues λn , λn , λn and real normalized eigenvectors ψn (i, α), ψn (i, α), ψn (i). The spin-dependent potential can be written as a sum of squares of single-particle operators as Vsd =

15A 1 X 2 λm O m , 2 m=1

(5)

where the 15A operators are given by X (σ) Onα = σiα ψn(σ) (i, α) , i

(στ ) Onαβ

=

X

τiα σiβ ψn(στ ) (i, β) ,

i

(τ ) Onα

=

X

τiα ψn(τ ) (i) ,

(6)

i

1

2

with n = 1, A. We apply the Hubbard–Stratonovich transformation e− 2 ∆tλO = √ R∞ x2 √1 dxe− 2 + −λ∆txO to linearize the quadratic dependence in the operators 2π −∞ Om . The variable x corresponds to an auxiliary field. Each of the 15A terms in Eq. (5) requires an auxiliary field. We write the short time approximation of the spin-dependent propagator as " 15A # Z X x2 p n −Vsd ∆t , (7) e = dX exp − − xn −λn ∆tOn 2 n=1

Q15A dxn where dX ≡ n=1 √ and we drop commutator terms which are higher order than 2π ∆t on the right hand side.

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Once the Hubbard–Stratonovich variables have been sampled, the resulting propagator acting on a walker (i.e. positions and spinors) gives a single new walker. x2 We importance sample the auxiliary field variables xn by first writing 2n + √ √ √ x2 xn −λn ∆tOn as 2n + xn −λn ∆thOn i + xn −λn ∆t(On − hOn i), where hOn i = hΨT |On |R, Si/hΨT |R, Si is the mixed expectation value, and then by keeping the first two terms to form a shifted contour Gaussian as in Ref. 44. The trial function hΨT |R, Si can be complex. Therefore the random walk must be constrained. To this aim we use, here, the fixed-phase approximation43,44 , which implies constraining hΨT |R0 , S 0 i to have the same phase of hΨT |R, Si, rather than requiring that the real part of their ratio be ≥ 0 as in the constrained path condition. Applying FP, the walker weight can be reexpressed in terms of the local energy EL (R, S) = Re(hΨT |H|RSi/hΨT |RSi) (see Ref. 50 for more details) Our algorithm becomes: i) sample |R, Si initial walkers from |hΨT |R, Si|2 using Metropolis Monte Carlo; ii) propagate in the usual DMC way with a drifted Gaussian for a time step; iii) diagonalize, for each walker, the potential matrices A(σ) , A(τ ) and A(στ ) ; iv) sample the corresponding shifted contour auxiliary field variables and update the spinors. The new walker has a weight given by exp(−EL (R0 , S 0 )∆t). It should be noted that the AFDMC algorithm can be numerically applied, without any particular extra effort to simulate either nuclear systems with N 6= Z 37 or deformed nuclei. 3. Open and Closed Shell Nuclei We present in this section the results obtained with FP–AFDMC for the binding energy of the nuclei 4 He, 8 He, 16 O and 40 Ca51 and compare them with the available results obtained with other many-body methods. The Jastrow function f c (r) used in the calculations is given by the first component f (1) (r) of the FHNC/SOC correlation operator which minimizes the energy per particle of symmetric nuclear matter at equilibrium density ρ = 0.16 f m−3 . Radial orbitals φi are of the type ~ CM , si ), where R ~ CM = PA ~ri /A is the center of mass of the nucleus. They φi (~ri − R i=1 have been calculated by using self-consistent Hartree–Fock approximation with the Skyrme’s effective interactions of Ref. 52, used to study light nuclei. Given a set of positions and spinors, the antisymmetrization produces a determinant of single particle orbitals. For open-shell nuclei, a sum of several determinants is used to build a trial wave function having a definite angular momentum J. Our trial function contains no tensor correlations and the variational estimate is not even bound. The diffusion process enforced by the AFDMC method is capable of crossing the transition from an unbound to a bound system, leading to energy estimates which compare very well with the available GFMC results as can be seen in Table 1. For the alpha particle FP–AFDMC estimates are compared with GFMC and the Effective Interaction Hyperspherical Harmonic (EIHH) methods. 53 The FP–AFDMC agreement with GFMC and EIHH for 4 He is within about 1% of the total energy. The agreement between AFDMC and GFMC for 8 He is even better.

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We have compared the FP–AFDMC results for 16 O with the available results of other methods. Variational FHNC/SOC,48 and Cluster Variational Monte Carlo (CVMC)47 have been used the Argonne v14 interaction to compute the ground state energy of 16 O. Our result for the energy, keeping just the first six operators of Argonne v14 , is -90.8(1) MeV. The CVMC result, when keeping just the same six operators, is -83.2 MeV, whereas -84.0 MeV has been obtained by FHNC/SOC calculations. Therefore, FP–AFDMC lowers the energy by about 10% with respect to the two different variational results. Calculations for the 16 O, 40 Ca and symmetric nuclear matter, at ρ0 with the Argonne v60 , have also been performed in order to ascertain (i) the scalability of the AFDMC method in the number of nucleons, and (ii) the degree of accuracy in the description of this semi-realistic interaction for closed shell nuclei. Results Table 1. Taken from Ref. 51. Ground-State energy of 4 He, 8 He, 16 O and 40 Ca for the Argonne v60 interaction computed with FP–AFDMC. Coulomb energies EC are not included. GFMC results are extracted from Ref. 46, assuming EC = 0.7M eV . EIHH results are from Ref. 54. Experimental energies are from Ref. 55. We also include the estimate of the energy of SNM at equilibrium density, simulated with 28 nucleons in a periodic box56 (see Sec. IV). All values are expressed in MeV. nucleus

EAF DM C

EGF M C

EEIHH

EAF DM C /A

Eexp /A

4 He

−27.13(10) −23.6(5) −100.7(4) −272(2)

−26.93(1) −23.6(1)

−26.85(2)

−6.78 −2.95 −6.29 −6.80 −12.8(1)

−7.07 −3.93 −7.98 −8.55 −16

8 He 16 O 40 Ca

SNM

are reported in Table 1, where it is reported for completeness also the energy of symmetric nuclear matter (SNM) at the equilibrium density ρ0 ,56 which provides the volume term in the Weizs¨ acker mass formula of v60 interaction. As expected the v60 interaction is not at all sufficient to build the binding energy of 16 O and of 40 Ca. It gives about 96% of the binding energy for alpha particle, 75% for 8 He, 79% for 16 O and 79% for 40 Ca. Our 16 O is unstable to break up into 4 alpha particles, and our 40 Ca has the same energy of 10 alpha particles. This behavior is consistent with the simple pair counting argument of Ref. 57. The surface energy coefficient in the Weizs¨ acker formula, resulting from the comparison of the binding energies per nucleon of symmetrical nuclear matter and 40 Ca is 20.5 MeV, not too far from the experimental value of 18.6 MeV. An important conclusion which can be drawn from the results presented in this section is that AFDMC has the same level of accuracy as GFMC, with the great advantage of allowing for efficient simulations of large nuclei. 4. Nuclear Matter The properties of nuclear matter, like the Equation of State (EOS), the excitation spectrum, the Green’s function and the momentum distribution, the response func-

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tions and the structure functions, are of fundamental importance in Nuclear Physics, mainly because nuclei behave very much like liquid drops.58 Indeed, each of these can be associated with a mass formula, which fits the corresponding data of stable nuclei from A ∼ 20 on. Any such mass formula has a volume and a symmetry term provided by symmetrical nuclear matter and nuclear matter with N > Z respectively. Moreover, accurate model independent calculations of the above observables are much needed in the physics of heavy ion reactions, as well as in that of lepton and neutrino scattering off nuclei at intermediate energies. Medium effects have to be taken into account for the data analyses of such reactions at the present level of accuracy. In addition, the theoretical knowledge of the properties of asymmetric nuclear matter at low temperature is needed to predict the structure, the dynamics and the evolution of stars, in particular during their last stages, when they become ultradense neutron stars. Depending on the EOS, the density of nuclear matter in the inner shells can reach up to 9 times its equilibrium density ρ0 .59 FHNC and FHNC/SOC calculations have marked a fundamental step in the understanding of the microscopic base of the liquid drop model and the shell effects, which provide only an explanation of the gross features of nuclei, having linked them directly to the NN interaction in the vacuum. In that, they have increased the accuracy of our theoretical estimates by at least an order of magnitude. However, we need a further step forward. For this, as mentioned in the introduction, a better knowledge of the nuclear forces at densities of the order or larger than ρ0 , and more accurate ways of solving the nuclear many-body Hamiltonian are required. The role played by relativistic corrections and that of nucleonic excitations or mesons, integrated out in most of the modern NN interactions, are also expected to be very important. Very little is known from the quantitative point of view about it. We put these problems in a second priority with respect to the first ones, although some of them might have a strong interplay with the understanding of nuclear forces at high density. At present, the main properties of nuclear matter, such as the equilibrium density, the binding and symmetry energy and the compressibility do not enter into the fits of the nuclear data to get the semi-phenomenological NN interactions. That is because the theoretical estimates of these quantities are still too poor to take them into account. The EOS calculated from first principles, directly from the modern nuclear interactions do not agree with the experimental one. It is not really known whether that is due to the importance of the neglected many-body forces or to the inaccuracy of the FHNC/BHF or other many-body methods of comparable accuracy. We present in the following the results obtained with FP–AFDMC for the EOS of symmetric nuclear matter56 and of pure neutron matter in normal phase, as well as the gap of the BCS phase of neutron matter.50,60

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4.1. Symmetric nuclear matter We have performed calculations of the binding energy of symmetric nuclear matter in correspondence of two different two-body NN potentials of the v6 -type: the first one is v60 and we denote the second one as v600 . The last potential is obtained by truncating the Argonne v80 46 at the v6 level, namely by dropping out the two spinorbit components. Argonne v80 , Argonne v60 has been obtained by projecting the Argonne v18 interaction to M = 8 level, so to describe the deuteron correctly. It follows that, differently from v60 , v600 does not reproduce the binding energy of the deuteron and cannot be considered a semi-realistic potential. The reasons for using these two potentials are the following. Our first priority has been that of working out the tensor and the tensor-τ components of the two-body interaction, which we considered the main source of inaccuracy of our original CP–AFDMC. We have been chosen v60 to apply AFDMC to a broad range of nucleonic systems from light nuclei up to nuclear matter with a semi-realistic interaction, containing tensor force and already used in previous light nuclei GFMC calculations or other calculations of comparable accuracy. The choice of v 600 enables a straightforward comparison with FHNC/SOC and/or BHF calculations of symmetric nuclear matter. As in the case of the simulations of nuclei, discussed in the previous section, the Jastrow function f c (r) has been extracted from a FHNC/SOC calculation of symmetric nuclear matter at the desired density ρ. We have calculated the binding energy with 28 nucleons in a periodic box for ten different values of the density in the range 0.5 ≤ ρ/ρ0 ≤ 3. The results of the calculations with A=28 include box Table 2. Taken from Ref. 56. AFDMC energies per particle in MeV of 28, 76 and 108 nucleons in a periodic box at various densities for the v600 potential. ρ/ρ0

E/A(28)

E/A(76)

E/A(108)

0.5 3.0

−7.64(3) −10.6(1)

−7.7(1) −10.7(6)

−7.45(2) −10.8(1)

corrections, computed by adding to the two body sums contribution of nucleons in the first shell of periodic cells, finding that such procedure is effective. In order to assess the magnitude of finite size effects we performed calculations with 76 and 108 nucleons at densities ρ = 0.08 fm−3 and ρ = 0.48 fm−3 . The results are shown in Table 2. As it can be seen, they coincide with the ones obtained with 28 nucleons within 3 percent. In the case of 28 nucleons for each density we generated and then propagated a set of 1000 walkers for different time-steps ranging from ∆t = 5 × 10−6 MeV−1 to ∆t = 2.5 × 10−5 MeV−1 . Each propagation at each time-step were performed up to at least a total imaginary time of t = 2MeV−1 . The AFDMC energy is determined by extrapolating to ∆t → 0. In order to lower statistical errors, in some case longer total propagation time was needed, up to a maximum of t = 6MeV−1

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-8

AFDMC fit AFDMC FHNC/SOC FHNC/SOC + elem. BHF

-10

E [MeV]

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-12

-14

-16 0.5

1

1.5

ρ /ρ0

2

2.5

3

Fig. 1. Taken from Ref. 56. EOS of symmetric nuclear matter calculated with different methods with the v600 potential. Solid green line with circles: our fixed-phase AFDMC results with statistical error bars; dashed line with squares: FHNC/SOC;49 dashed line with diamonds: BHF.49 The triangles correspond to the FHNC/SOC energies corrected by including the lowest order of elementary diagrams as described in the text.

in particular at higher densities. Using a parallel supercomputer (typically 16 CPU are employed) a propagation of 20000 steps requires about 80 processor hours. Then for a fixed density we estimated that a maximum of 5000 CPU hours are needed. In the case of 76 and 108 nucleons we performed calculations only at a one time-step ∆t = 10−5 MeV−1 and we propagated until a total imaginary time of t = 1MeV−1 . The statistical error has been kept below five per thousand at all the densities considered. The EOS of symmetric nuclear matter obtained with 28 nucleons for the v600 potential is displayed in Fig. 1, where it is also compared with results obtained using both FHNC in SOC approximation and BHF in the two-hole line approximation.49 If we assume that our AFDMC estimates suffer very little by final size effects, as suggested by the results given in Table 2, and if their accuracy is as good as that reached for 4 He and 8 He nuclei, then Fig. 1 deserves the following comments: (i) FHNC/SOC leads to an overbinding, particularly at high density. A similar indication was found by Moroni et al.61 after a Diffusion Monte Carlo calculation of the EOS of normal liquid 3 He at zero temperature, with a guiding function including triplet and backflow correlations. The comparison was made with the corresponding FHNC/SOC calculations of Refs. 62 and 63 with a correlation operator having the scalar and the ~σi · ~σj components. It is very difficult to assess the degree of accuracy of the approximations introduced in FHNC theory by the neglecting of the elementary diagrams and of terms arising from the non commutativity of the correlation operators, intrinsic of the FHNC/SOC approximation. It results that such approximations violate the variational principle. The triangles in Fig. 1 show that the lowest order elementary diagram, namely the one having only one correlation

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bond and four exchange bonds, gives a sizable contribution, bringing, presumably by mere chance, the FHNC/SOC in very close agreement with FP–AFDMC. (ii) BHF calculations of Ref. 49 predict an EOS with a shallower binding than the AFDMC one. It has been shown for symmetric nuclear matter, using Argonne v18 , that contributions from three hole-line diagrams add a repulsive contribution up to ∼ 3MeV at densities below ρ0 ,64 and decrease the energy at high densities.65 Such corrections, if computed with Argonne v60 potential, would probably preserve the same general behavior, bringing the BHF EOS closer to the AFDMC one. Therefore, our calculations show that the two hole-line approximation used in Ref. 49 is too poor, particularly at high density. (iii) As expected, the saturation density ρs provided by the v600 model is too high (ρs = 1.83ρ0 , with a binding energy per particle of 14.04(4) MeV). It is well known that three-body force, with its repulsive contribution, increasing with density, is crucial to bring ρs closer to ρ0 . However, it will also raise the energies of the EOS. Therefore, even though the binding energy at ρ0 , which is 11.5(1) MeV, is 1.3 MeV smaller than that provided by a semi-realistic force like v60 (see Table 1), it seems unlikely that a realistic interaction such as Argonne v18 plus Urbana IX threebody will reproduce the experimental EOS. If our guess is correct, then m-body forces with m > 3 will be important even at ρ0 , and therefore they will be crucial for getting the EOS and pressure vs. density estimates at the theoretical accuracy required today.

4.2. Pure neutron matter Extensive CP–AFDMC calculations have been performed for PNM36–41 with the Argonne v80 two-body potential plus the Urbana IX three-nucleon interaction. In the following we present and discuss preliminary results50,60 obtained for both the normal and BCS phases with FP–AFDMC using the same two– plus three-body interaction. A comparison of the CP–AFDMC results with those obtained with FHNC/SOC66 and BHF67 methods and GFMC simulations of 14 neutrons,68 in spite of an overall agreement, shows the following discrepancies: (i) the CP–AFDMC EOS are more repulsive of the FHNC/SOC ones; (ii) an important source of discrepancy comes from the contribution from spin-orbit interaction, which results to be much smaller in CP–AFDMC calculations. The use of spin-orbit induced backflow correlations in the trial function has reduced the discrepancy in a marginal way only; (iii) the gap energies of the BCS-phase are significantly larger than those obtained in previous calculations. Fig. 2 displays the new FP–AFDMC results for the EOS of the normal phase of PNM, obtained with v80 alone and by adding also the Urbana IX three-nucleon force, and compares them with the corresponding CP–AFDMC ones. One can see a significant lowering of the energy per particle going from the CP to the FP constraint. In particular, the contribution from spin–orbit interaction comes out to be

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34 60 AFDMC-FP, v8’+UIX AFDMC-FP, v8’ AFDMC-CP, v8’+UIX AFDMC-CP, v8’

50

40 E [MeV]

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30

20

10 0.1

0.2

0.15

0.25

0.3

-3

ρ [fm ]

Fig. 2. Comparison of the CP–AFDMC and FP–AFDMC results for normal PNM with and without Urbana IX three-nucleon force. The results shown are all obtained with 66 neutrons. Finite size effects are not included.

in much better agreement with the previous calculations. However, the comparison with the results by Akmal et al.66 still shows that FHNC/SOC leads to a too soft EOS. The differences are larger when three-body interaction is switched on, particularly at high density. It is worth observing how important is the three-nucleon interaction already at medium-high densities. Its contribution at 2ρ0 is ∼ 25M eV and increases very rapidly with density. The four Illinois potentials,27 built to include two ∆ intermediate states in the three nucleon processes, lead to very different results compared to the Urbana IX37 EOS at medium-high densities, in spite of the fact that all of them provide a satisfactory fit to the ground state and the low energy spectrum of nuclei with A ≤ 8. This, once more, points outs the importance of understanding the role of many-body forces in nuclear astrophysics. We have Table 3. Comparison of the FP–AFDMC gap energies of PNM in BCS phase calculated with the CP–AFDMC ones.39 Calculations are performed with v80 plus Urbana IX potential and N=64-68 neutrons. The energies are in MeV. The CP result at k F = 0.4f m−1 has been obtained with the v80 potential and using N=12-18 neutrons. kF (f m−1 ) 0.4 0.6

∆F P

∆CP

∆F P /EF

1.5(2) 2.1(5)

1.8(1)∗

0.74 0.47

2.8(1)

also repeated60 the calculations of the gap energies of the BCS phase of PNM at various densities, using the FP constraint. Some preliminary results are reported in Table 3. The gaps at kF = 0.4f m−1 and kF = 0.6f m−1 are compared with the corresponding ones by Fabrocini et al.39 calculated with CP–AFDMC. All the results, except for the CP one at kF = 0.4f m−1 , have been obtained with v80 plus Urbana IX three-nucleon potential, using N=64-68 neutrons in a periodic box. The CP case

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100

0

0.2

0.4

0.6

0.8

-3

ρ [fm ]

Fig. 3. The FP–AFDMC equations of state of PNM, with and without the inclusion of Urbana IX three-nucleon force, are compared with the FHNC/SOC66 ones. All the results shown have been obtained with 66 neutrons. Finite size effects are not included.

at kF = 0.4f m−1 has been runned with three-body potential switched off and using N=12-18 neutrons. Above kF = 0.8f m−1 and below kF = 0.1f m−1 FP–AFDMC with 66 neutrons finds that E(BCS) is marginally larger than E(normal). In conclusion, the above preliminary results indicate that the use of the FP constraint lower the CP gaps of about 20-30 percent and does not find any BCS superfluidity outside the range kF ∼ [0.1 − 0.8]f m−1 . 5. Conclusion and Perspectives We reported on the state of art of AFDMC calculations on nucleonic systems, with particular attention to nuclei and symmetric nuclear matter, for which the tensor-τ interaction plays a major role. It has been shown that the use of the fixed phase constraint allows for a better sampling of the random walks compared to the constrained path one, leading to an overall lowering of the energies. AFDMC has proved to have reached, in its FP formulation, the same level of accuracy of GFMC and the most powerful few-body techniques in calculating the binding energies of light nuclei both in the closed and open configurations. However, better than these methods, it can be applied to calculating, at unprecedented accuracy and with no extra difficulties from asymmetries or deformations, heavy nuclei and nuclear matter. Nuclei, up to A=40 nucleons, and nuclear matter, simulated by up to 108 nucleons in a periodic box, have been calculated with semi-realistic two-body potentials containing tensor and tensor-τ forces. The computational time per imaginary time step scales as A3 . The total computational time depends on the desired quantity, the distribution of excited states, and the quality and complexity of the trial function just as in all other quantum Monte Carlo methods.

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Moreover, the results obtained, even if performed with semi-realistic interactions, strongly indicate that many-body forces are going to play a fundamental role in the understanding of nuclear matter properties, particularly at medium and high densities, a region of great interest in nuclear astrophysics and heavy ion physics. We believe that AFDMC, particularly in the new FP formulation, has opened up a number of important frontiers in nuclear physics and nuclear astrophysics, some of which are already subjects of intense exploration. From the methodological point of view, the following problems are in first priority: (i) inclusion of the missing v18 components; (ii) inclusion of Urbana IX and Illinois I–IV three-body potentials in the calculations of nuclei and nuclear matter; (iii) use of twist–averaged boundary conditions69,70 in AFDMC; (iv) full development of periodic box FHNC theory 71,72 for nuclear matter in normal and superfluid phase. Most of them are already at a quite advanced stage of development. In particular we want to discuss in some more details the advancements we are making in the treatment of three-body force, mainly because it may be linked to the exploration of many-body force models. The Urbana IX potential is obtained integrating out the pions and the Delta degrees of freedom in the three-nucleon process having a Delta intermediate state, under the assumption of infinite masses for the nucleons and the Delta. This integration gives rise to two terms, the anticommutator one, which is easy, because it can be reduced to a two-body spin-isospin operator, and a commutator one, which is zero in PNM and is difficult, because it leads to a three-body spin-isospin operator.37 Such a term needs double Hubbard–Stratonovich integration and therefore gives large variance. Including an explicit Delta with finite mass reduces the variance. An interesting possibility that we are exploring is that of using a fictitious Delta having a single particle energy K = p2 /2m + ∆m, where m is the usual nucleon mass, and the same spin-isospin of the nucleon. The advantage of using this rather than the small momentum expansion of the relativistic energy is that the Gaussian part of the Green’s function will not change. It is easy to verify that in the limit of ∆m → ∞ one can get back the Urbana IX potential. Besides being an efficient way to compute three-body force, the above procedure will allow an exploration of a finite fictitious mass ∆m, as a way of including many-body forces at all orders, so as to reproduce the experimental EOS of symmetric nuclear matter. The following calculations are in progress: (i) the full EOS of nuclear matter at zero temperature, as a function of density and the asymmetry (N − Z)/A with realistic interaction, like v18 plus Urbana IX (or possibly better than that) and its implications on the structural and rotational properties of neutron stars; (ii) pion condensation in nuclear matter; (iii) closed shell nuclei with realistic interactions.

Acknowledgments We wish to thank A.Sarsa, A.Yu Illarionov, S.Vitiello, S. Pieper, R.B. Wiringa and J. Carlson for having provided us with important information and for the illuminating

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discussions. This work was supported in part by NSF grant PHY-0456609 and by INFN, Sezione di Trieste. Most of the calculations were performed on the HPC facility “BEN” at ECT* in Trento under a grant for Supercomputing Projects, on the HPC facility WIGLAF at the Physics Department of the University of Trento, and on the HPC facility “E–Lab” at SISSA. References 1. V. R. Pandharipande, I. Sick and P. W. Huberts, Rev. Mod. Phys. 69, p. 981 (1997). 2. O. Benhar, A. Fabrocini, S. Fantoni, G.A.Miller, V. R. Pandharipande and I. Sick, Phys. Rev. C 44, p. 2328 (1991). 3. O. Benhar, A. Fabrocini, S. Fantoni, V. R. Pandharipande and I. Sick, Phys. Rev. Lett. 69, p. 881 (1992). 4. O. Benhar, N. Farina, H. Nakamura, N. Sakuda and R. Seki, Phys. Rev. D 72, p. 053005 (2005). 5. G. G. Raffelt, The stars as laboratories of fundamental physics (University of Chicago, Chicago & London, 1996). 6. R. F. Sawyer, Phys. Rev. C 40, p. 865 (1989). 7. N. Iwamoto and C. J. Pethick, Phys. Rev. D 25, p. 313 (1982). 8. S. Reddy, M. Prakash, J. M. Lattimer and J. Pons, Phys. Rev. C 59, p. 2888 (1989). 9. S. Fantoni and S. Rosati, Nuovo Cimento A 20, 179 (1974). 10. S. Fantoni and S. Rosati, Nuovo Cimento A 25, p. 593 (1975). 11. E. Krotcheck and M. L. Ristig, Nucl. Phys. A 242, p. 389 (1975). 12. V. R. Pandharipande and R. B. Wiringa, Rev. Mod. Phys. 51, p. 821 (1979). 13. E. Feenberg, Theory of Quantum Fuids (Academic Press, New York, 1969). 14. J.W.Clark, Progr. in Part. and Nucl. Phys. 2, 89 (1979). 15. A. D. Jackson, E. Krotscheck, D. E. Meltzer and R. A. Smith, Nucl. Phys. A 386, p. 125 (1982). 16. S. Fantoni, Phys. Rev. B 29, p. 2544 (1984). 17. S. Fantoni and V. R. Pandharipande, Phys. Rev. C 37, p. 1697 (1988). 18. S. Fantoni and A. Fabrocini, Lecture Notes in Phys. 510, 119 (1998). 19. S. Fantoni, B. L. Friman and V. R. Pandharipande, Nucl. Phys. A 386, p. 1 (1982). 20. S. Fantoni, Nuovo Cimento A 44, 191 (1978). 21. S. Fantoni and V. R. Pandharipande, Nucl. Phys. A 427, 473 (1984). 22. O. Benhar, A. Fabrocini and S. Fantoni, Nucl. Phys. A 505, 267 (1989). 23. O. Benhar, A. Fabrocini and S. Fantoni, Nucl. Phys. A 550, 201 (1992). 24. I. Sick, S. Fantoni, A. Fabrocini and O. Benhar, Phys. Lett. B 333, p. 267 (1994). 25. A. Fabrocini and S. Fantoni, Nucl. Phys. A 503, p. 375 (1989). 26. B. S. Pudliner, V. R. Pandharipande, J. Carlson, S. C. Pieper and R. B. Wiringa, Phys. Rev. C 56, 1720 (1997). 27. S. C. Pieper, V. R. Pandharipande, R. B. Wiringa and J. Carlson, Phys. Rev. C 64, p. 014001 (2001). 28. S. C. Pieper, K. Varga and R. B. Wiringa, Phys. Rev. C 66, p. 044310 (2002). 29. B. Borasoy, E. Epelbaum, H. Krebs, D. Lee and U.-G. Meissner, Eur. Phys. J. A 31, p. 105 (2007). 30. D. R. Entem and R. Machleidt, Phys. Rev. C 68, p. 041001(R) (2003). 31. K. E. Schmidt and M. H. Kalos, Monte Carlo Methods in Statistical Phys. , 125 (1984). 32. M. A. Lee, K. E. Schmidt, M. H. Kalos and G. V. Chester, Phys. Rev. Lett. 46, p. 728 (1981). 33. S. C. Pieper, Nucl. Phys. A 751, p. 516 (2005). 34. K. E. Schmidt and S. Fantoni, Phys. Lett. B 446, p. 99 (1999).

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35. S. Zhang, J. Carlson and J. Gubernatis, Phys. Rev. Lett. 74, p. 3652 (1995). 36. S. Fantoni, A. Sarsa and K. E. Schmidt, Phys. Rev. Lett. 87, p. 181101 (2001). 37. A. Sarsa, S. Fantoni, K. E. Schmidt and F. Pederiva, Phys. Rev. C 68, p. 024308 (2003). 38. L.Brualla, S.Fantoni, A.Sarsa, A. Fabrocini, K. E. Schmidt and S. Vitiello, Phys. Rev. C 67, p. 065806 (2003). 39. Adelchi Fabrocini, Stefano Fantoni, Alexei Yu. Illarionov and K. E. Schmidt, Phys. Rev. Lett. 95, p. 192501 (2005). 40. F. Pederiva, A. Sarsa, K. E. Schmidt and S. Fantoni, Nucl. Phys. A 742, 255 (2004). 41. S. Gandolfi, F. Pederiva, S. Fantoni and K. E. Schmidt, Phys. Rev. C 73, p. 044304 (2006). 42. K. Schmidt, S. Fantoni and A. Sarsa, Eur. Phys. J. A 17, p. 469 (2003). 43. G. Ortiz, D. M. Ceperley and R. M. Martin, Phys. Rev. Lett. 71, 2777 (1993). 44. S. Zhang and H. Krakauer, Phys. Rev. Lett. 90, p. 136401 (2003). 45. R. B. Wiringa, V. G. J. Stoks and R. Schiavilla, Phys. Rev. C 51, p. 38 (1995). 46. R. B. Wiringa and S. C. Pieper, Phys. Rev. Lett. 89, p. 182501 (2002). 47. S. C. Pieper, R. B. Wiringa and V. R. Pandharipande, Physical Rev. C 46, 1741 (1992). 48. A. Fabrocini, F. Arias de Saavedra and G. Co’, Phys. Rev. C 61, p. 044302 (2000). 49. I. Bombaci, A. Fabrocini, A. Polls and I. Vida˜ na, Phys. Lett. B 609, p. 232 (2005). 50. S. Gandolfi, PhD thesis (2007). 51. S. Gandolfi, F. Pederiva, S. Fantoni and K. E. Schmidt, Phys. Rev. Lett. 99, p. 022507 (2007). 52. X. Bai and J. Hu, Phys. Rev. C 56, p. 1410 (1997). 53. N. Barnea, W. Leidemann and G. Orlandini, Phys. Rev. C 61, p. 054001 (2000). 54. G. Orlandini, private communication (2006). 55. Table of nuclides hhttp://atom.kaeri.re.kri, (2000). 56. S. Gandolfi, F. Pederiva, S. Fantoni and K. E. Schmidt, Phys. Rev. Lett. 98, p. 102503 (2007). 57. R. B. Wiringa, Phys. Rev. C 73, p. 034317 (2006). 58. A. Bohr and B. Mottelson, Nuclear Structure (Benjamin, New York, 1969). 59. J. Piekarewicz, Phys. Rev. C 69, p. 041301 (2004). 60. S. Gandolfi, A. Y. Illarionov, F. Pederiva, S. Fantoni and K. E. Schmidt, in preparation (2007). 61. S. Moroni, S. Fantoni and G. Senatore, Phys. Rev. B 52, p. 13547 (1995). 62. E. Manousakis, S. Fantoni, V. R. Pandharipande and Q. N. Usmani, Phys. Rev. B 28, p. 3770 (1983). 63. M. Viviani, E. Buendia, S. Fantoni and S. Rosati, Phys. Rev. B 38, p. 4523 (1988). 64. H. Q. Song, M. Baldo, G. Giansiracusa and U. Lombardo, Phys. Rev. Lett. 81, p. 1584 (1998). 65. M. Baldo, A. Fiasconaro, H. Q. Song, G. Giansiracusa and U. Lombardo, Phys. Rev. C 65, p. 017303 (2001). 66. A. Akmal, V. R. Pandharipande and D. G. Ravenhall, Phys. Rev. C 58, p. 1804 (1998). 67. M. Baldo, U. L. G. Giansiracusa and H. Q. Song, Phys. Lett. B 473, p. 1 (2000). 68. J. Carlson, J. Morales, Jr., V. R. Pandharipande and D. G. Ravenhall, Phys. Rev. C 68, p. 025802 (2003). 69. C. Lin, F. H. Zong and D. M. Ceperley, Phys. Rev. E 64, p. 016702 (2001). 70. A. Y. Illarionov, private communication (2007). 71. S. Fantoni and K. E. Schmidt, Nucl. Phys. A 690, 456 (2001). 72. F. A. de Saavedra, private communication (2007).

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STATIC AND DYNAMIC MANY-BODY CORRELATIONS E. KROTSCHECK† and C. E. CAMPBELL†‡ † Institut ‡ School

f¨ ur Theoretische Physik, Johannes-Kepler-Universit¨ at, A-4040 Linz, Austria

of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455

This paper presents a systematic development of the linear equations of motion for a dynamically correlated wave function. In the past, only time-dependent pair correlations have been treated in a reasonably consistent manner. We argue here that this level is insufficient to describe many physical effects in the energy/momentum regime of the roton, and that at least time-dependent three-body correlations are necessary to capture the relevant physics. For the specific problem of atomic impurities in 4 He, we develop, for the first time, the theory to the level of time-dependent triplet fluctuations. The need for a symmetric self-energy operator and the desire to formulate the theory in terms of quantities that have a clear diagrammatic definition suggests the introduction of a very specific set of physical variables. The very plausible final result says that, to leading order, the “Feynman” spectra in the energy denominators are replaced by self-consistent spectra that are expressed in terms of the self-energy. We expect that a very similar structure will emerge in the self-energy of pure 4 He.

1. Introduction The most successful semi-analytic method for describing the ground state of strongly correlated liquids such as 4 He is undoubtedly the Jastrow–Feenberg method that starts with a specific hierarchy of correlated ground state wave functions and calculates ground state properties after high-order diagram summations and optimization. One of the reasons for this success is that the theory sums, in a very specific “local” approximation, the so-called “parquet” class of diagrams.1 More importantly it also suggests specific truncation schemes that lead to numerically large cancelations, but are not at all obvious from the point of view of Green’s function theories.2 To describe dynamics, it is therefore only natural to generalize the Jastrow– Feenberg theory to time-dependent wave functions. The approach goes back to work by Feenberg, Jackson, and Campbell.3–5 In this paper, we bring the theoretical description of the dynamics of 4 He to a new level by including time dependent triplet correlations. We formulate the method here for an atomic impurity; this was first done by Owen6 in the long-wavelength limit and later generalized to finite wavelengths, cf. Ref. 7. There are two reasons that we formulate the theory for the perhaps slightly mundane and well studied

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problem of impurities in 4 He instead of the bulk liquid. One is that the theoretical description is somewhat simpler because it involves up to five-body distribution functions, whereas a theory for the bulk liquid would, at the same level, need sixbody distribution functions. The second is a recent experimental interest in the damping of the motion of muonic helium in 4 He. Especially for this problem, quantitative estimates of the energy loss of a muonic impurity due to coupling to background excitations are needed. We will show that our description addresses exactly this problem. While the calculations are somewhat tedious, our results will be sufficiently simple such that the implementation of the same theory for the bulk liquid seems quite feasible. To argue why present theoretical descriptions of impurity motion in 4 He are insufficient to include damping, let us look at the kinematics of a 3 He impurity in 4 He. Fig. 1 shown the experimental8 phonon-roton spectrum in 4 He, the “Feynman spectrum” eF (k) = ~2 k 2 /[2m4 S(k)], the kinetic energy t3 (k) = ~2 k 2 /[2m3 ] of a free 3 He impurity, and the kinetic energy of a 3 He impurity with an effective mass of m∗3 = 2.15 m3 which is the experimental value at zero pressure.9

30 25 20 hω [K]

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15 e(k) 10

eF(k) h2k2 2m3 h2k2 2m*3

5 0 0.0

0.5

1.0

1.5

2.0 2.5 q (Å−1)

3.0

3.5

4.0

Fig. 1. The figure shows, for zero pressure, the experimental8 phonon-roton spectrum in 4 He (solid line with error bars), “Feynman spectrum” eF (k) = ~2 k 2 /[2m4 S(k)] (long-dashed line), the kinetic energy t3 (k) = ~2 k 2 /[2m3 ] of a free 3 He impurity (short-dashed line), and the kinetic energy of a 3 He impurity with an effective mass of m∗3 = 2.15 m3 which is the experimental value at zero pressure9 (dotted line).

Damping can occur whenever the 3 He impurity can lose its energy by coupling to the phonon-roton spectrum. As we will see, the presently best theory still treats

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the background excitations at the level of Feynman’s theory of excitations. It is evident that such a simple description leads to a quantitatively incorrect result; in particular it would predict that, once the 3 He atom acquires an effective mass above 2m3 , the impurity motion cannot be damped. The problem was repaired in the past by simply using the experimental phonon-roton spectrum as an input to the theory; that approach is questionable because it would put all of the strength of the 4 He excitation into the collective mode, whereas it is well known that there is a large amount of multi-particle excitations in the roton regime where damping becomes relevant. Technically, we write the time-dependent wave function as Φ(t) = q

1 ΨI0

e I

I −iEN +1 t/~

| Ψ0

ΨI0 (r0 , r1 , ...rN ; t) ,

(1)

where ψ I (r0 , r1 , ...rN ; t) contains the time-dependent correlations, written in a Jastrow–Feenberg form (2) ΨI0 (r0 , r1 , ...rN ; t) = exp 21 δU (r0 , r1 , ...rN ; t) ΨIN +1 (r0 , r1 , ..., rN ) , X X δu2 (r0 , ri , rj , ; t) . (3) δu2 (r0 , ri ; t) + δU (r0 , . . . , rN ; t) = δu1 (r0 ; t) + i

i<j

I 4 EN +1 is the ground state energy of N-body liquid He with one impurity particle I and ΨN +1 is the corresponding ground state wave function. The coordinate r0 will always refer to the impurity, also specifying the kind of correlation or density under consideration. The ground state is assumed to be exact. It can, without loss of generality, be written in Jastrow–Feenberg form; we will point out where the assumption of a Jastrow-form is necessary. The time-dependent correlations are determined by stationarity of the action integral Z Z ∂ I S = dtL(t) = dt hΦ(t)| HN |Φ(t)i , (4) +1 − i~ ∂t I where HN +1 is the Hamiltonian of the system containing N background atoms and one impurity. Since we are concerned with weak perturbations, we expand the Lagrangian to second order:

I I 1 I

I Lint (t) = ΨI0 (t) HN ΨN +1 [δU ∗ , [T, δU ]] ΨIN +1 +1 − EN +1 Ψ0 (t) = 8 X ~2

2 = ΨIN +1 |∇i U | ΨIN +1 (5) 8mi i

I ∂ I i~ I ∗ ˙ ∗ I ˙ Lt (t) = Ψ0 (t) i~ Ψ0 (t) = ΨN +1 δU δ U − δU δ U ΨN +1 . (6) ∂t 8

The linearized equations of motion are then

δ [Lint (t) − Lt (t)] 1 (EOM)i (r0 , ..ri−1 ; t) = = 0, 4 δ(δu∗i (r0 , ..ri−1 ; t))

(7)

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where, in particular, i~ δLt (t) = ρ˙ i (r0 , . . . , ri−1 ; t) . δ(δu∗i (r0 , ..ri−1 ; t)) 4(i − 1)!

(8)

The ρ˙ i (r0 , . . . , ri−1 ; t) are understood to be complex functions; the physical density fluctuation is the real part. The ρi (r0 , . . . , ri−1 ; t) are the (time-dependent) i-body densities. For further reference, we will also need the i-body distribution functions gi (r0 , . . . , ri−1 ; t) =

ρi (r0 , . . . , ri−1 ; t) . ρ1 (r0 ) . . . ρ1 (ri−1 )

(9)

2. Triplet Fluctuations 2.1. Densities and distribution functions The key step in the execution of the theory when including triplet fluctuations is the introduction of the new fluctuating variables. The calculations become rather lengthy, and we can show only the most essential manipulations and steps. For brevity, also define d3 ρi ≡ ρ1 (ri )d3 ri . We will demonstrate that the most useful variables are related to, but not identical with, the fluctuating density and distribution functions. The physical fluctuating density corresponding to the above wave function can be expressed in terms of n-body ground state densities. The physical density fluctuation is the real part of the generally complex function " Z δρ1 (r0 ; t) ≡ ρ1 (r0 ) δu1 (r0 ; t) + 1 + 2

Z

3

3

d3 ρ1 g2 (r0 , r1 )δu2 (r0 , r1 ; t) #

d ρ1 d ρ2 g3 (r0 , r1 , r2 )δu3 (r0 , r1 , r2 ; t) .

(10)

This suggests the introduction of δv1 (r0 ; t) = δρ1 (r0 ; t)/ρ1 (r0 ) instead of δu1 (r0 ; t) as the fluctuating one-body variable. Introducing a useful two-body variable is the key step. We can formally write Z δg2 (r0 , r1 ) δg2 (r0 , r1 ) = d3 r2 δu2 (r0 , r2 ; t) δu2 (r0 , r2 ) Z δg2 (r0 , r1 ) + d 3 r2 d 3 r3 δu3 (r0 , r2 , r3 ; t) (11) δu3 (r0 , r2 , r3 ) and define, for further reference, (I)

1 δg2 (r0 , r2 ) 1 δg2 (r0 , r1 ) = (12) ρ1 (r2 ) δu2 (r0 , r2 ) ρ1 (r1 ) δu2 (r0 , r1 ) δ(r1 − r2 ) = g2 (r0 , r1 ) + [g3 (r0 , r1 , r2 ) − g2 (r0 , r1 )g2 (r0 , r2 )] . ρ1 (r1 )

F2 (r0 ; r1 , r2 ) ≡

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Define now a new function δv2 (r0 , r2 ; t) through Z (I) δg2 (r0 , r1 ; t) = d3 ρ2 F2 (r0 ; r1 , r2 )δv2 (r0 , r2 ; t) , Z δv2 (r0 , r2 ; t) = δu2 (r0 , r2 ; t) + d3 ρ3 d3 ρ4 G(r0 ; r2 , r3 , r4 )δu3 (r0 , r3 , r4 ; t), (13) which defines the four-point function G(r0 ; r2 , r3 , r4 ). To interpret this quantity, go back to Eq. (11): Z δg2 (r0 , r1 ) (I) = d3 ρ4 F2 (r0 , r1 , r4 )G(r0 , r4 , r2 , r3 )ρ(r2 )ρ(r3 ) . (14) δu3 (r0 , r2 , r3 ) On the other hand, we have the symmetry δg2 (r0 , r1 ) 1 δg3 (r0 , r2 , r3 ) 1 = . ρ(r2 )ρ(r3 ) δu3 (r0 , r2 , r3 ) 2ρ(r1 ) δu2 (r0 , r1 )

(15)

Solving Eq. (14) for G(r0 ; r1 , r2 , r3 ) therefore leads to the relationship G(r0 , r1 , r2 , r3 ) =

1 δg3 (r0 , r2 , r3 ) . 2ρ(r1 ) δg2 (r0 , r1 )

(16)

This formula is useful for deriving the diagrammatic expansion of G(r0 , r1 , r2 , r3 ). A few generic examples contributing to G(r0 , r1 , r2 , r3 ) are shown in Fig. 2.

3

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Fig. 2. The graphical representation of a few diagrams of the four-point function G(r 0 , r1 , r2 , r3 ). A δ(r1 − r2 )/ρ1 (r1 ) function is implied in the two cases where points r1 and r2 coincide. The first three diagrams are included in the convolution approximation.

2.2. Current and time-derivative term The usefulness of the new two-body variable is demonstrated for two quantities. It is evident that there is a one-to-one connection between δv2 (r0 , r; t) and δg2 (r0 , r; t), thus we could also have used δg2 (r0 , r; t). However, this would mean that we would

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have to calculate the inverse of the operator F2 (r0 ; r1 , r2 ). While this is not a prob(I) lem, F2 (r0 ; r1 , r2 ) is defined entirely in terms of distribution functions, whereas its inverse is not. First, let us look at the particle current:

I I ΨN +1 ˆj (r0 )δU ΨIN +1

I j1 (r0 ; t) ≡ ΨN +1 | ΨIN +1 Z h ~ ρ1 (r0 ) ∇0 δu1 (r0 ; t) + d3 ρ1 g2 (r0 , r1 )∇0 δu2 (r0 , r1 ; t) = 2mI i Z i 1 + d3 ρ1 d3 ρ2 g3 (r0 , r1 , r2 )∇0 δu3 (r0 , r1 , r2 ; t) , (17) 2

where ˆjI (r0 ) is the particle current operator for the impurity. In terms of the new variables, the one-body current (17) becomes Z h ~ ρ1 (r0 ) ∇0 δv1 (r0 ; t) − d3 ρ1 δv2 (r0 , r1 ; t)∇0 g2 (r0 , r1 ) j1 (r0 ; t) = 2mI i Z 1 − d3 ρ1 d3 ρ2 δu3 (r0 , r1 , r2 ; t)∇0 g3 (r0 , r1 , r2 ) 2 Z i + d3 ρ1 d3 ρ2 d2 ρ3 δu3 (r0 , r1 , r2 ; t)G(r0 , r3 , r1 , r2 )∇0 g2 (r0 , r3 ) . (18)

In the last two lines, we now use the variational definition (16). Doing the derivative with respect to point r0 differentiates all g2 (r0 , ri )-bonds. We can therefore write, for a Jastrow wave function for which g3 (r0 , r1 , r2 ) can be expanded exactly in terms of g2 (ri , rj ) bonds, Z δg3 (r0 , r1 , r2 ) ∇0 g3 (r0 , r1 , r2 ) = d3 ρ3 ∇0 g(r0 , r3 ) δg(r0 , r3 ) Z = 2 d3 ρ3 ∇0 g(r0 , r3 )G(r0 , r3 , r1 , r2 ) .

In other words, the last two terms in Eq. (18) cancel for Jastrow wave functions. Only if the ground state contains triplet correlations does a non-trivial term remain. As a second important quantity, we look at the time-derivative part of the Lagrangian. Starting with Eqs. (8) and using Eqs. (10) and (13), we get

i~ δLt (t) = ρ(r ˙ 0 ; t) (19) δ(δv1 (r0 ; t)) 4 i~ δLt (t) = [ρ˙ 2 (r0 , r1 ; t) − ρ˙ 1 (r0 ; t)ρ1 (r1 )g2 (r0 , r1 )] δ(δv2 (r0 , r1 ; t)) 4 i~ = ρ1 (r0 ; t)ρ1 (r1 )g˙ 2 (r0 , r1 ; t) (20) 4 δLt (t) i~ = g˙ 3 (r0 , r1 , r2 ; t) − (21) δ(δu∗3 (r0 , r1 , r2 ; t)) 8 Z 2 d3 ρ4 g˙ 2 (r0 , r4 ; t)G(r0 , r4 , r1 , r2 ) ρ1 (r0 )ρ1 (r1 )ρ1 (r2 ).

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The time-dependence of the three-body distribution function comes from the timedependence of the pair fluctuations and the time dependence of the triplet fluctuations. The pair fluctuations have been re-expressed in terms of the function δv2 (r0 , ri ; t) which has a one-to-one relationship to the pair distribution function. We can therefore write Z δg3 (r0 , r1 , r2 ) g˙ 3 (r0 , r1 , r2 ; t) = d3 ρ3 g˙ 2 (r0 , r3 ; t) δg2 (r0 , r1 ) Z δg3 (r0 , r1 , r2 ) + d3 ρ3 d3 ρ4 δ u˙ 3 (r0 , r3 , r4 ; t) . (22) δu3 (r0 , r3 , r4 )

The first term cancels exactly against the G term. Thus, the result is that, in these variables, the time-dependence of the thee-body term comes solely from the timedependence of the three-body fluctuations: Z i~ 3 δLt (t) δg3 (r0 , r1 , r2 ) = ρ1 (r0 )ρ1 (r1 )ρ1 (r2 ). d ρ3 d3 ρ4 δ u˙ 3 (r0 , r3 , r4 ; t) δ(δu∗3 (r0 , r1 , r2 ; t)) 8 δu3 (r0 , r3 , r4 ) 2.3. Interaction Lagrangian The most tedious task is to rewrite the interaction Lagrangian (5) in terms of the new variables. To avoid excessively long equations, we split the interaction Lagrangian into an “impurity” piece that contains all gradients wrt. r0 and a “background” piece that contains all gradients wrt. r1 . We skip the technical details of the derivation. In the end, we can rewrite the interaction part of the Lagrangian as Z mI (I) (B) 2 d3 ρ0 |v(t)| + Lint (t) + Lint (t) (23) Lint (t) = 2 where v(r0 ; t) ≡ j(r0 ; t)/ρ(r0 ) is the velocity field, and Z ~2 (I) (I) d3 ρ0 d2 ρ1 d3 ρ2 [∇0 δv2∗ − δu∗3 (∇0 G)] F2 · [∇0 δv2 − (∇0 G)δu3 ] Lint (t) = 8mI Z ~2 (I) + d3 ρ0 . . . d3 ρ4 (∇0 δu∗3 )F3 (∇0 δu3 ) (24) 16mI Z ~2 (B) (B) Lint (t) = d3 ρ0 d2 ρ1 d3 ρ2 [∇1 δv2∗ − δu∗3 (∇1 G)] F2 · [∇1 δv2 − (∇1 G)δu3 ] 8mB Z ~2 (B) d3 ρ0 . . . d3 ρ4 (∇1 δu∗3 )F3 (∇1 δu3 ) , (25) + 8mI (B)

where F2 ≡ g2 (r0 , r1 ) has been introduced for symmetry of notation. The func(I) (B) tions F3 and F3 are combinations of up to five body distribution functions. 3. Equations of Motion 3.1. General form Without being specific about the approximations we will eventually use in particular (I) (B) for the operators G, F3 , and F3 , we can write down the equations of motion. For

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that purpose, assume harmonic time dependence of all time-dependent variables, which simply means that we can everywhere substitute i~∂/∂t → ~ω. The one-body equation is simply the continuity equation ∇0 j(r0 , t) + δ ρ(r ˙ 0 , t) = 0

(26)

where j(r0 , t) is understood to be given by the first two terms in Eq. (18), i.e. it is a functional of δv1 (r0 ; t) and δv2 (r0 , r1 ; t). Doing the variation wrt. δu∗3 (r0 , r1 , r2 ) and δv2∗ (r0 , r1 ) gives equations of the formal structure T33 (ω)δu3 + T32 δv2 = 0 ,

T22 (ω)δv2 + T23 δu3 = i~j(r0 ; ω) · ∇0 g2 (r0 , r1 ) .

(27) (28)

(I,B)

The operators Tii (ω) come from the Fi combined with the time derivative terms, and the T32 from the coupling terms. We can formally eliminate the triplet fluctuations and get a new two-body equation: −1 T22 (ω) − T23 T33 (ω)T32 δv2 ≡ T (ω)δv2 = i~j(r0 , ω) · ∇0 g2 (r0 , r1 ) . (29) Thus, the only consequence of triplet fluctuations is a redefinition of the operator T (ω) which is, when triplet fluctuations are ignored, simply T22 (ω). In the case of a homogeneous system, the one-body variable is a plane wave, δv1 (r0 ) = v1 eiq·r0 and we can also separate δv2 (r0 , r1 ) = eiq·r0 vq(2) (r1 − r0 ),

δu3 (r0 , r1 ) = eiq·r0 u(3) q (r1 − r0 , r2 − r0 ) . (30)

Going to momentum space, we make the distinction between background and impurity quantities by superscripts (I) and (B). The one-body equation (26) becomes 2 2 Z ~2 d3 p ~ q ~2 q 2 (I) (2) (I) ˜ δv1 + q · ph (p)˜ vq (p) ≡ + Σ (q, ω) δv1 = ~ωδv1 . 2mI 2mI (2π)3 ρ 2mI (31) Inserting the formal solution of Eq. (29), we can define the self-energy as Σ(I) (q, ω) = [1 + I(q, ω)] Σ(I) u (q, ω) , 2 2 Z 3 d p1 d3 p2 ~ ˜ (I) (p1 )Te −1 (p1 , p2 )q · p2 ˜ q · p1 h h(I) (p2 ) (32) Σ(I) q u (q, ω) = 2mI (2π)6 ρ2

where T˜q (p1 , p2 ) is the momentum space representation of the T (ω), spelling out the variables more explicitly, and I(q, ω) will be defined in Eq. (33). The inverse is understood in the sense of the convolution product. Eq. (32) defines the “unrenormalized” self energy, which leads to Owen’s “unrenormalized” effective mass.6 On the other hand, we have Z 2 2 ~2 d3 p ˜ u(2) (p) ≡ ~ q I(q, ω)v1 Σ(I) (q, ω)v1 = q · ph(p)˜ (33) q 3 2mI (2π) ρ 2mI

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and hence (I)

Σ(I) (q, ω) =

Σu (q, ω) 1−

2mI (I) ~2 q 2 Σu (q, ω)

.

(34) (I)

The final form of the dispersion relation, expressed in terms of Σu (q, ω), is then

1−

~2 q 2 2mI 2mI (I) 2 2 ~ q Σu (q, ω)

= ~ω .

(35)

3.2. Pair excitation limit To make our point, let us go back to the pair approximation which neglects (I) δu3 (r0 , r1 , r2 ). The only open question is then how to deal with the operators F2 (B) (B) (I) and F2 . The “uniform limit approximation” sets F2 = 1 and F2 (r0 , r1 , r2 ) ≈ S(r1 , r2 ), where S(r, r0 ) =

δ(r − r0 ) + h(r, r0 ) ρ(r)

is the coordinate space static structure function of the background. In this approximation, one obtains a plausible answer 2 2 2 Z ˜ (I) (p) 3 · p X k d p ~ S(p) . Σ(I) u (k, ω) = 2mI (2π)3 ρ ~ω − tI (k − p) − F (p)

(36)

(37)

˜ (I) (k) is the “direct correlawhere S(k) is the 4 He static structure function, X 2 2 tion function,” tI (k) = ~ k /2mI is the impurity kinetic energy, and F (p) = ~2 k 2 /(2mB S(k)) is the Feynman spectrum of the background system. We will refer to this approximation as the “unrenormalized self-energy”. For completeness and further reference, we will also need, in the same approximation, the background density-density response function: χ(B) (k, ω) =

S(k) S(k) + , ~ω − F (k) − Σ(k, ω) −~ω − F (k) − Σ(k, −ω)

(38)

with (neglecting triplet correlations) a self-energy 5 Σ(B) (k, ω) =

1 2

2

~ 2mB

2 Z

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d pd q δ(k + p + q) (2π)3 ρ

2 ˜ ˜ · p X(p) + q X(q) k S(p)S(q)

. ~ω − F (p) − F (q) (39) Equation (37) clearly exhibits the problems of the “pair excitation” approximation mentioned above. In the energy denominator, we have a bare impurity energy, and a Feynman excitation spectrum that lies, in the roton region, more than a factor of two above the experiment. Thus, one cannot expect that a coupling of the 3 He motion to the 4 He background is described correctly. Clearly, we have made approximations in the development of the above “uniform limit” approximation for the self-energy. However, removing these approximations S(k)

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cannot be the cure of the problem: The reason for that is that the lowering of the roton minimum is achieved by permitting short-ranged fluctuations between two impurity atoms. Such fluctuations are not included in the excitation operator (3) as long as only one background coordinate is retained. Thus, while it is computationally (I) quite feasible10 to use better approximations for F2 (r0 , r1 , r2 ), the uniform limit approximation displays the content of the theory more clearly.

3.3. Triplet fluctuations in the uniform limit approximation The equations of motion become quite complicated at the three-body level, and a diagrammatic analysis to pick out the most important terms is necessary. Without being very precise about what is meant by the “uniform limit approximation,” we use the term as approximation for all terms that contain triplet correlations that can be written entirely as products in momentum space. In that approximation, we obtain (I)

Z ~2 d3 ρ0 d3 ρ1 d3 ρ2 ∇0 δv2∗ (r0 , r1 ) · ∇0 δv2∗ (r0 , r2 )S(r1 , r2 ) (40) 8mI Z ~2 − d3 ρ0 d3 ρ1 d3 ρ2 d3 ρ3 ∇0 δv2∗ (r0 , r1 ; t) · ∇0 h(r0 , r2 ) 8mI × δu3 (r0 , r3 , r2 ; t)S(r1 , r3 ) + c.c. Z ~2 d3 ρ0 d3 ρ1 d2 ρ2 d3 ρ3 d3 ρ4 ∇0 δu∗3 (r0 , r1 , r2 ) · ∇0 δu3 (r0 , r3 , r4 ) + 16mI × S(r1 , r3 )S(r2 , r4 ) , Z ~2 2 ≈ d3 ρ0 d3 ρ1 |∇1 δv2∗ (r0 , r1 )| (41) 8mB Z ~2 d3 ρ0 d3 ρ1 d3 ρ2 d3 ρ3 ∇1 δv2∗ (r0 , r1 ; t) · ∇1 h(r1 , r2 ) − 2 ~ m8B × δu3 (r0 , r3 , r2 ; t)S(r1 , r3 ) + c.c. Z ~2 + d3 ρ0 d3 ρ1 d3 ρ2 d3 ρ3 S(r2 , r3 )∇1 δu∗3 (r0 , r1 , r2 ; t) · ∇1 δu3 (r0 , r1 , r3 ; t) . 8mB

Lint ≈

(B)

Lint

The only non-trivial time-derivative term is the three-body function. Consistent with the “uniform-limit approximation,” the time-derivative term is δLt (t) i~ = ∗ δ(δu3 (r0 , r1 , r2 ; t)) 8

Z

d3 ρ3 d3 ρ4 δ u˙ 3 (r0 , r3 , r4 ; t)S(r3 , r1 )S(r4 , r2 ) .

(42)

In the uniform limit approximation and in momentum space, we simply have T22 (q0 , q1 ) = S(q1 ) [tI (q0 ) + F (q1 ) − ~ω]

T33 (q0 , q1 , q2 ) = S(q1 )S(q2 ) [tI (q0 ) + F (q1 ) + F (q2 ) − ~ω] .

(43) (44)

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With the ansatz (30), the three-body equation of motion becomes δ˜ u(3) q (p1 , p2 ) =

~2 2mB

−

~2 2mI

(45)

(2) δ˜ vq (p1

˜ (B) (p1 ) + p2 X ˜ (B) (p2 )) + p2 )(p1 + p2 ) · (p1 X tI (q + p1 + p2 ) + F (p1 ) + F (p2 ) − ~ω (2) (2) ˜ (I) (p2 )δ˜ ˜ (I) (p1 )δ˜ (q + p1 ) · p2 X vq (p1 ) + (q + p2 ) · p1 X vq (p2 ) tI (q + p1 + p2 ) + F (p1 ) + F (p2 ) − ~ω

and the correction term [T23 δu3 ]q (p) becomes Z ~2 d3 p0 0 ˜ (I) (p0 )S(p)˜ [T23 δu3 ]q (p) = u(3) (q + p) · p0 h q (p, p ) 2mI (2π)3 ρ Z ~2 d3 p0 0 0 ˜ (B) (p0 )S(p − p0 )˜ − u(3) p · p0 h q (p − p , p ). (46) 2mB (2π)3 ρ Inserting the solution (45) of the three-body equation into Eq. (46) leads to a number of terms. Some of them are non-local in the sense that they involve integrations over (2) the p dependence of δuq (p). But two terms, among those the one that contains a (2) factor (~2 /2mB )2 , are local in the sense that they contain an explicit factor δvq (p). These “local” terms can be identified with the bulk and the impurity self-energy: loc. [T23 δu3 ]q (p) = h i (p + q, ~ω − (p)) . − S(p)δ˜ vq(2) (p) Σ(B) (p, ~ω − tI (p + q)) + Σ(I) F u

(47)

We hesitate at this point to claim that the omission of the non-local terms is generally permitted; however, the local terms show quite nicely what has been accomplished: Instead of the simple energy denominator (43), we obtain a “self-consistent” energy denominator T (q0 , q1 ) = S(q1 ) tI (q0 ) + Σ(I) u (q0 , ~ω − F (q1 )) (B) + F (q1 ) + Σ (q1 , ~ω − tI (q0 )) − ~ω . (48) Thus, the message is very simple: Including triplet fluctuations renormalizes the energy denominator in the self-energy (37) by self-energy corrections. The result is quite plausible apart from the fact that the theory gives the energies and momenta for which these corrections have to be calculated. 4. Numerical Applications One attractive feature of our result is that most of the bits and pieces of the calculation can be taken from existing theoretical calculations. In our first applications, we disregard the “non-local” terms and work in the “uniform limit approximation.” It is not a priori clear whether any of these approximations are quantitatively legitimate. In fact, although some of the approximations used in our previous work7 can

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be relaxed today, that work gives indications that the inclusion of the full operators (I,B) F2 (r0 , r1 , r2 ) leads to quantitative corrections to the effective mass of the order of 10 to 20 percent. These results are, of course, not entirely microscopic since, observing that the energy denominator in Eq. (32) contains the Feynman spectrum and not the correct phonon-roton spectrum, the experimental spectrum was put in “by hand.” This procedure must definitely be checked very carefully because it basically assumes that the background spectrum is, even at high momenta, a single collective mode. Concerning fluctuating triplets, we will demonstrate that the “local” formulation captures already the right physics. Figure 3 show our results for the imaginary part of the 3 He single-particle propagator, 1 (49) =m ~ω − tI (q) − Σ(I) (q, ω) for four different calculations. The upper two panes show calculations using the pairfluctuation approximation and the “renormalized self-energy” (34) as well as the (I) “unrenormalized” form Σ(I) (q, ω) ≈ Σu (k, ω). The lower two panes show the same, now including triplet fluctuations. Also shown is a rough effective mass fit of the spectrum in the regime 0 ≤ k ≤ 1 ˚ A−1 and, for the pair excitation approximation, the Feynman spectrum and for the triplet excitation theory the CBF spectrum resulting from the response function (38). For the calculations, we have added a small imaginary part of width 0.05 K to the energy to broaden the spectrum in those regions where it is a δ-function. A number of observations can be made: The most prominent difference originating from triplet fluctuations is the onset of damping. Expectedly, that onset is determined, in the pair-fluctuation approximation, by the location of the roton minimum in the Feynman approximation, whereas it is determined by the location of the CBF roton minimum when triplet fluctuations are included. This is not only expected, but also the effect we are looking for. The second observation is that the long-wavelength excitation spectrum is practically unaltered by triplet fluctuations. Both calculations lead to practically the same spectrum. The “unrenormalized” theory predicts an effective mass ratio of about 3, which is far too large, whereas the “renormalized” theory leads to an average effective mass of about 1.7, which falls short of the experimental value by about 20 percent. We must attribute this to the “uniform limit approximation.” It is not a big problem to include an improved version of the operator F2 (r0 , r1 , r2 ) that is consistent for both long- and short wavelengths; this will be done in future work.10 From a practical point of view, we note that the background spectrum derived from the response function (38) is still somewhat too high, i.e. while our best calculation (lower right pane of Fig. 3) is qualitatively correct when compared to the experimental curves in Fig. 1 — with damping of the impurity spectrum occurring at wave numbers just below the experimental roton — our spectra are still at somewhat too high energy. We doubt that this is a consequence of the uniform

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Fig. 3. The figure shows gray-scale plots for different calculations of the single-particle propagator (49) of a 3 He atom in 4 He. Darker areas correspond to higher values. The upper two panes show the calculation omitting triplet fluctuations, whereas the lower two panes show the calculation including triplet fluctuations. The left two panes show the “unrenormalized” and the right two panes the “renormalized” approximation. Also shown are effective mass fits to the long-wavelength spectrum (solid lines), the Feynman spectrum of the background (dashed lines, upper two panes) and the CBF spectrum of the background (dashed lines, lower two panes).

limit approximation, rather one suspects that the energy denominators in Eq. (32) get further renormalized by higher-order multi-particle fluctuations. We have in the past corrected for this deficiency by scaling the energy denominator in Eq. (38) appropriately. This procedure would certainly be legitimate here because one would be simulating the effect of yet higher order fluctuations. The concern mentioned above that an ad-hoc replacement of the Feynman spectrum by the experimental phonon-roton spectrum would put too much weight into the collective mode while neglecting multi-particle excitations does not apply. We have refrained from such ad-hoc modifications in this paper simply to keep the calculation free from phenomenological input. 5. Summary In this paper we have reported on the first formulation of the equations of motion for time-dependent three-body fluctuations. We have argued this is motivated by the fact that even a complete pair-fluctuation theory does not capture the right

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physics in the regime where the impurity motion is damped. We have utilized the “uniform limit approximation” which might occasionally look rather brutal. Some justification for that can be drawn from the fact that this approximation did quite well already at the pair fluctuation level, but of course some of the consequences could and should be tested. On the other hand, we have derived a very plausible new form of the impurity self-energy, in which simply the energy denominator gets renormalized. The resulting form is simple enough that the prospects for formulating the same methodology for the background system are quite promising. One immediate outcome of that would be the roton-roton coupling which causes the “plateau” in the experimental phonon-roton spectrum. This will be done in forthcoming work. Acknowledgments We would like to thank Mikko Saarela for lively discussions on the subject matter, and David Taqqu on alerting us to the experimental interest in the problem of muonic helium impurities. References 1. A. D. Jackson, A. Lande and R. A. Smith, Physics Reports 86, 55 (1982). 2. V. Apaja, J. Halinen, V. Halonen, E. Krotscheck and M. Saarela, Phys. Rev. B 55, 12925 (1997). 3. H. W. Jackson and E. Feenberg, Ann. Phys. (NY) 15, 266 (1961). 4. H. W. Jackson and E. Feenberg, Rev. Mod. Phys. 34, 686 (1962). 5. C. C. Chang and C. E. Campbell, Phys. Rev. B 13, 3779 (1976). 6. J. C. Owen, Phys. Rev. B 23, 5815 (1981). 7. E. Krotscheck, J. Paaso, M. Saarela, K. Sch¨ orkhuber and R. Zillich, Phys. Rev. B 58, p. 12282 (1998). 8. R. A. Cowley and A. D. B. Woods, Can. J. Phys. 49, 177 (1971). 9. S. Yorozu, H. Fukuyama and H. Ishimoto, Phys. Rev. B 48, 9660 (1993). 10. C. E. Campbell and E. Krotscheck, (2007), in preparation.

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ENTANGLEMENT IN MANY-BODY QUANTUM PHYSICS FRANK VERSTRAETE Faculty of Physics, University of Vienna The study of entanglement in quantum many-body systems has opened up new ways of understanding the nature of the wavefunctions encountered in strongly correlated quantum systems. We will briefly review some recent developments in that field, and discuss concepts like area-laws, numerical renormalization group methods and matrix product states. Keywords: Entanglement; renormalization group; DMRG; matrix product states; projected entangled pair states.

One of the most defining events in physics during the last decade has been the spectacular advance made in the field of strongly correlated quantum many body systems: the observation of quantum phase transitions in optical lattices and the realization that many-body entanglement can be exploited to build quantum computers are only two of the notable breakthroughs. In a remarkable turn of events, the tools developed in the context of quantum information science have been shown to shed a new light on the ones used to describe strongly correlated quantum manybody systems as studied in a wide variety of fields and has opened up many exciting research avenues. This is precisely the subject of this small writing. The key ingredient that distinguishes the quantum from the classical world is the concept of entanglement. As a response to the EPR-paper in 1935,1 Schr¨ odinger coined the concept of entanglement2 and immediately recognized it as being the defining characteristic of quantum mechanics. In the early-days of quantum mechanics however, people were too busy with the many successful applications of quantum mechanics to really pay attention to such foundational issues. Things changed drastically in the 60’s when John Bell, working as a high energy physicist in CERN, made the discovery that distributed entangled quantum states can in principle exhibit correlations that are stronger than correlations allowed for by local hidden variable models.3 Although a loophole-free Bell experiment has still not been performed, Bell’s work anticipated the fascinating quest to contrast the power of quantum versus classical information processing and was one of the main catalysts for the exceptional progress made in experimental quantum optics during the last decades. As a next logical step, visionary people like D. Deutsch, C. Bennett and P. Shor understood that entanglement can be exploited to do information tasks such

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as computing and cryptography much more efficiently than possible in a classical world. The current effort in the field of quantum information science is aimed at realizing those ideas and deepening our insights of the quantum vs. classical question. Obviously, this led to an explosion of work on entanglement theory. How do you define entanglement? In the words of Schr¨ odinger, a pure quantum state is entangled iff the whole is more than the sum of its parts. More specifically, the Hilbert space of a many-body system (where many ≥ 2) is a tensor product of local Hilbert spaces (those can correspond to e.g. modes in Fock space, to localized spins, to the polarization of a photon, ...). A pure quantum state is called separable if and only if the global wavefunction is a product of such single-particle wavefunctions, and entangled otherwise; a mixed quantum state is called entangled iff it cannot be written as a convex sum of pure separable states. Note that the possibility of entanglement is nothing more than a direct consequence of the superposition principle. Note also that the notion of entanglement strongly depends on the choice of local Hilbert spaces: a Slater determinant state is considered unentangled with relation to its normal mode decomposition, but can be highly entangled from the local point of view if these modes are delocalized. A lot of work has been done to quantify entanglement. In the case of a bipartite pure quantum system |ψAB i, the natural measure of entanglement is the von-Neumann entropy of the local reduced density operator ρA or ρB : S(|ψAB i) = −TrρA log2 (ρA ) = −TrρB log2 (ρB ). In essence, this von-Neumann entropy quantifies the maximal amount of Shannon information that A can obtain by doing a measurement on his part about the measurement outcome of part B. An entropic criterion is obviously desirable as the amount of entanglement of two copies of the same state is then equal to 2 times the original entanglement, but there is also a deeper reason why this measure is used: it can be proven that any collection of states with a given mean entanglement can be interconverted into any other collection of states with the same mean entropy by only local operations and classical communication.4 The von-Neumann entropy is therefore the unique measure that quantifies how useful entanglement is from the local point of view. Clearly, entanglement appears everywhere in quantum mechanical systems, and there are many complementary viewpoints on it. From the point of view of quantum information theory, it is a resource that allows for revolutionary information theoretic tasks such as quantum computation and quantum cryptography. Indeed, if not much entanglement would be generated during a quantum computation, it would be possible to simulate it on a classical computer. From the point of view of quantum many-body physics, entanglement gives rise to quantum phase transitions and exotic new phases of matter exhibiting e.g. topological quantum order. From the point of view of the numerical simulation of strongly correlated quantum systems such as quantum spin systems and quantum chemistry, entanglement is the

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enemy number one as it makes simulation so hard. Of course, these viewpoints are mutually compatible: the complexity of simulating entangled quantum systems is intimately connected to the power of quantum versus classical computation; the possibility of topological quantum order turns out to be strongly related to the notion of quantum error correction. It is this interplay between those complementary viewpoints that makes the study of entanglement such a rich subject. Recently, there has been much interest in investigating the amount and type of entanglement that is naturally present in strongly correlated quantum systems. On the one hand, this was motivated by the question of whether the amount and type of entanglement needed to do quantum computation could be present in the ground-state wave functions of quantum spin systems. On the other hand, the hope was that the study of entanglement in strongly correlated quantum systems could elucidate the underlying structure of the associated wavefunctions, which on its turn might lead to new ways of simulating them. Concerning the first question, a local 5-body quantum spin 1/2 Hamiltonian on a square lattice was identified whose ground state is a so-called cluster state and allows for universal quantum computation by doing adaptive local single-qubit measurements on it.5,6 This was a surprising result as it showed that ground states of local 2-D quantum spin models contain enough entanglement for doing universal quantum computation (note that local one-qubit operations can never create entanglement). The associated Hamiltonian is unusual for a quantum Hamiltonian as it consists of a sum of local commuting terms; this means e.g. that local perturbations will never spread by virtue of Hamiltonian evolution. It turns out that these cluster states and the way to do quantum computation with them can be understood within the formalism of valence bond states or projected entangled pair states (PEPS). 6 This class of states turns out to be very relevant in the context of simulation of quantum spin systems, and we will later come back to them. Concerning the other question, we would like to get a better understanding on the nature of the wavefunctions present in ground states of strongly correlated quantum systems. The study of correlations, both quantum and classical, is an incredibly rich field and lies at the heart of many of the most exciting discoveries in the fields of statistical physics and quantum information theory: quantum phase transitions occur due to the appearance of long-range correlations, and the theory of entanglement is all about quantifying the amount of quantum correlations. Quoting Schr¨ odinger, a physical system exhibiting strong correlations is such that ”the whole is more than the sum of its parts” and is as such fundamental for the understanding of collective phenomena. The natural choice to quantify correlations is to look at the connected correlation functions hOA OB i − hOA ihOB i as a function of the distance between the regions A and B. Non-critical systems exhibit exponentially decaying correlation functions, leading to the definition of a correlations length. Despite the basic nature of this result, it has only been proven

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B wB wA

A [

Fig. 1. The lattice is divided into two regions, A and B, with borders ∂A and ∂B, respectively. The entanglement between the two regions A and B scales like the area ∂A as opposed to the volume.

very recently;7 the main technical ingredient to that proof was a so-called LiebRobinson bound on the velocity by which correlations spread with respect to local Hamiltonian evolution;8 Hastings used this to provide a local approximation to the negative energy, or annihilation, part of an operator in a gapped system.7 The notion of a correlation length is very fundamental and quantifies the amount of degree of localization of the relevant degrees of freedom in the system. Intuitively, the notion of a correlation length should set the length scale at which ”the whole becomes equal to the sum of its parts”; in other words, if the distance between A and B becomes much longer than the correlation length, we should have ρAB ' ρA ⊗ ρB . However, just looking at 2-point correlation functions can be problematic: there exist quantum states ρAB for which all two-point correlation functions are arbitrarily small, and nevertheless they can be proven to be very far from product states.9 In the case of zero-temperature quantum systems, another obvious choice for quantifying the amount of correlations present is to calculate the entanglement entropy of a region A of spins as a function of the size and shape of the region. It has been observed that the entanglement entropy for noncritical systems typically obeys an area law, i.e. scales as the size of the boundary |∂A|, indicating that the only correlations that are relevant are the ones around that boundary. 10 It has furthermore been shown that this area law is violated mildly for critical onedimensional quantum spin and critical two-dimensional free fermionic systems, for which a multiplicative logarithmic correction has to be added:11 S(ρA ) ' |∂A| log(|∂A|). However, there are also critical quantum spin systems known in 2 dimensions for which a strict area law holds,12 and hence it is a very interesting open question of how to relate area laws to the notion of a correlation length.

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In the case of non-zero temperature systems, correlations can be quantified by using the concept of mutual information: I(A : B) = S(ρA ) + S(ρB ) − S(ρAB ). The mutual information is zero if and only if the state is a product state ρA ⊗ ρB , has the same operational meaning as the entanglement entropy, and reduces to twice the entanglement entropy for pure states. Very recently, a bound on this mutual information, valid both for classical and quantum thermal states of local Hamiltonians of the form ρAB = exp(−βH)/Tr exp(−βH), has been derived.13 The argument works equally well for spin systems, for systems with infinite dimensional local Hilbert spaces such as bosons, and for fermionic systems. The proof is short enough to reproduce here. We consider two regions A and B, and write H as a sum of 3 terms HA , HB , H∂ acting respectively only on A, B and across the boundary. Thermal states are variationally characterized by the fact that they minimize the free energy; hence, any other state has a higher free energy, in particular ρ A ⊗ ρB with ρA , ρB the reduced density operators of the global thermal state ρAB . We hence have: Tr (HρAB ) − TS(ρAB ) ≤ Tr (HρA ⊗ ρB ) − TS(ρA ⊗ ρB ) which is equivalent to kH∂ k 1 Tr (H∂ ρA ⊗ ρB ) ≤ . T T The right hand side obviously scales like the boundary of region A as opposed to its volume, and this proves that any classical and quantum thermal state exhibits an exact area law at any non-zero temperature, even in the critical case. Another intriguing open problem is to connect this behaviour to the zero-temperature behaviour where logarithmic corrections arise in the case of integrable critical quantum spin chains. In summary, correlations are mainly concentrated around the boundary and the entropy of a block of spins scales like its boundary as opposed to its volume. That means that there is very few entanglement in ground states of local quantum Hamiltonians. This can be exploited to come up with a variational class of wavefunctions that captures this behaviour and is still easy to simulate, and this has precisely been the program that has been successfully pursued during the last years. In the case of 1-dimensional quantum spin systems, powerful numerical renormalization group (NRG) algorithms have been devised in the 70’s14 by Wilson ∗ ; those methods were later generalized to the density matrix renormalization group (DMRG) by S. White which allows to simulate ground state properties of any spin chain.15 Both of those methods have been extremely successful and allow to calculate correlation functions of the related systems up to almost machine precision. I(A : B) ≤

∗ Note

that Wilson mapped the Kondo impurity problem to an effective 1-D model as only s-wave electrons are scattered.

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d

D

¦¦ D

P

D D i j k cijk

1 i , j ,k 1

D

¦i

i

i 1

Fig. 2. A PEPS can be defined on any lattice; the small circles correspond to virtual Ddimensional spins, and the map P maps them to a physical spin of dimension d represented by the bigger ellipsoids.

Only recently, it has become clear that both of those renormalization algorithms can be rephrased as variational methods within the class of so-called matrix product states (MPS).16 The class of MPS is very much related to the valence-bond AKLT models put forward by Affleck, Kennedy, Lieb and Tasaki17 and the generalizations thereof known as finitely correlated states.18 MPS can also be generalized to socalled projected entangled pair states (PEPS) which can be defined on any lattice in any dimensions.19 How can we represent those MPS and PEPS? First of all, consider a graph where the physical d-dimensional spins lie on the nodes of the graph, and a collection of P virtual bipartite entangled EPR-pairs D i=1 (|ii|ii distributed along all vertices of the graph (see Fig. 2). Next, we want to identify a local subspace of those virtual spins to the space of physical spins by applying a linear map P to them that maps c spins (c being the coordination number of the graph) to one (physical) spin of dimension d. The class of PEPS is now obtained by letting those projectors P vary over all possible D c × d matrices. The AKLT-state is of that form, in which 2 qubits are mapped to a spin 1 state in the case of the 1-dimensional spin chain and 3 qubits to a spin 3/2 state in the case of a hexagonal lattice. The cluster state discussed earlier is also of that form, and the quantum computation going on when doing local measurements can be completely understood by identifying the underlying virtual qubits as the logical qubits.6 In the case of a 1-dimensional structure, those states are MPS, and an important aspect of them is that all correlation functions can be calculated with a computational cost that scales linearly in the number of spins and

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polynomially in D: although we work in an exponentially large Hilbert space, the class of MPS forms a subclass for which we can calculate all properties efficiently. By making use of entanglement theory, it has recently become clear why the numerical renormalization group methods are so successful: this is a consequence of the fact that this class of MPS is rich enough to approximate any ground state of a local gapped Hamiltonian efficiently. This implies that we have identified the manifold of ground states of all local gapped one-dimensional Hamiltonians: ground states are well approximated by MPS, and conversely all MPS are guaranteed to be ground states of local Hamiltonians. Similar statements hold for PEPS in higher dimensions. To be more precise, let’s define what we mean by good approximations. PN −1 Consider a family of Hamiltonians HN = i=1 hi,i+1 parameterized by the number of spins N and nearest neighbour interactions hi,i+1 , and associated ground states D |ψN i. The goal is to represent |ψN i by a MPS |ψN i such that D k|ψN i − |ψN ik ≤ .

The central question is: how does D has to scale as a function of 1/ and N such that this relation is fulfilled? If the scaling of D is polynomial in 1/ and N , then it means that the ground state is represented efficiently by a MPS: indeed, the previous equation implies that the expectation value of any observable on the exact ground state is arbitrary close to the one of the MPS, that the cost of getting a better precision does only scales polynomial † , and that all correlation functions can D be calculated efficiently on |ψN i. The previous requirements can be met under pretty broad assumptions. First of all, it has been proven that whenever an area law is satisfied for the exact ground state, a MPS will indeed exist that approximates it well with polynomial scaling in 1/, N .20 This argument works whenever the Renyi entropy Sα (ρ) = Tr(ρα )/(1 − α) for an α < 1 of a contiguous block of spins is bounded above by a constant times the logarithm of the size of the block. This turns out to be true for all spin chains for which this quantity has been calculated exactly, including the critical Heisenberg spin chain and the Ising Hamiltonian in a transverse magnetic field. In a related recent development, it has been proven that all gapped local Hamiltonians on a spin chain obey a strict area law in terms of the Renyi entropies21 implying the approximability by MPS. The central technical tool used in the proof is again the Lieb-Robinson bound.8 This provides a clear theoretical justification for the numerical renormalization group methods. There is however one caveat: it is not because the ground state can always be approximated very well with a MPS, that the computational complexity of actually finding it is also polynomial. It has indeed been proven that this problem is NP-complete and hence intractable in the worst case scenario.22 However, this does not have any implications for the difficulty of † Note

that it is otherwise trivial to simulate any quantum spin system with a cost polynomial in N and exponential in 1/: divide the spin system in blocks, neglect the boundary terms connecting those blocks, and find the ground state for all blocks independently.

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simulating physical quantum spin chains, as nature itself is not believed to be able to solve NP-complete problems: the situations in which simulation with MPS fail seem to be exactly the ones for which nature cannot relax to its ground state. In recent years, there has been an explosion of work showing how the formalism of MPS/PEPS can be used to simulate ground states of quantum spin systems. These methods, especially the ones in 2 dimensions, are still in development, but it has become clear that they offer crucial new insights into the structure of the wavefunctions to be found in nature. Although originally formulated on the level of quantum spin systems, it has become clear that they can be used in a much broader context. Consider e.g. fermionic lattice spin systems. This is of particular interest as we would like to identify the phase diagram for the 2-dimensional Hubbard model. Fortunately, it has recently been proven that local Hamiltonians of fermions can always be mapped to local Hamiltonians of spins,23 independent of the dimension, and therefore the whole formalism of PEPS is equally applicable to fermionic lattice systems. Those methods can equally well be used to simulate quantum field theories, non-equilibrium systems, classical spin systems, and work is under way to tackle problems in quantum chemistry. In conclusion, we have argued that entanglement theory provides fundamental new insights into the nature of strongly correlated quantum spin systems. It turns out that the amount of entanglement in ground states of quantum spin systems is surprisingly low. Under pretty general assumptions, this has allowed us to identify the manifold of wavefunctions associated to the low-energy sector of strongly correlated quantum spin systems, which on its turn can be applied to devise powerful ab-initio numerical variational methods for simulating them.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47,777 (1935). E. Schr¨ odinger, Proc. Camb. Phil. Soc. 31,555, (1935). J.S. Bell, Physics 1,195, (1964). C.H. Bennett, H.J. Bernstein, S. Popescu, and B. Schumacher, Phys. Rev. A 53, 2046 (1996). R. Raussendorf and H.J. Briegel, Phys. Rev. Lett. 86, 5188 (2001). F. Verstraete and J.I. Cirac, Phys. Rev. A. 70, 060302(R) (2004). M.B. Hastings, Phys. Rev. Lett. 93, 140402 (2004). E.H. Lieb, D.W. Robinson, Commun. Math. Phys. 28, 251 (1972); M.B. Hastings, T. Koma, Commun. Math. Phys. 265, 781 (2006). P. Hayden, D. Leung, P. W. Shor, A. Winter, Commun. Math. Phys. 250, 371 (2004). P. Calabrese, J. Cardy, J. Stat. Mech. P06002 (2004); M.B. Plenio et al., Phys. Rev. Lett. 94, 060503 (2005). G. Vidal et al., Phys. Rev. Lett. 90, 227902 (2003); M.M. Wolf, Phys. Rev. Lett. 96, 010404 (2006). F. Verstraete, M. M. Wolf, D. Perez-Garcia, J. I. Cirac, Phys. Rev. Lett. 96, 220601 (2006). M.M. Wolf, F. Verstraete, M.B. Hastings, J.I. Cirac, arXiv:0704.3906. K.G. Wilson, Rev. Mod. Phys. 47, 773 (1975).

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15. S. White, Phys. Rev. Lett. 69, 2863 (1992); U. Schollw¨ ock, Rev. Mod. Phys. 77, 259 (2005). 16. F. Verstraete, D. Porras and J.I. Cirac, Phys. Rev. Lett. 93, 227205 (2004). 17. I. Affleck et al., Commun. Math. Phys. 115, 477 (1988). 18. M. Fannes, B. Nachtergaele and R. F. Werner, Comm. Math. Phys. 144, 443 (1992). 19. F. Verstraete and J.I. Cirac, cond-mat/0407066; V. Murg, F. Verstraete, J.I. Cirac, cond-mat/0501493. 20. F. Verstraete, J.I. Cirac, Phys. Rev. B 73, 094423 (2006). 21. M.B. Hastings, JSTAT, P08024 (2007). 22. N. Schuch, J.I. Cirac, F. Verstraete, in preparation. 23. F. Verstraete, J.I. Cirac, J. Stat. Mech., P09012 (2005).

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NEW STATES OF QUANTUM MATTER GORDON BAYM Department of Physics, University of Illinois Urbana, IL 61801, USA In recent years physicists have created new states of quantum matter — from quarkgluon plasmas in ultrarelativistic heavy ion collisions to new Bose and Fermi superfluids in ultracold trapped atomic clouds. Although these systems differ in energy scales by some 20 orders of magnitude, they share many questions of physics in common. Here we review these new states and discuss possible connections between the two areas. Keywords: Ultracold atomic gases; quark-gluon plasmas; ultrarelativistic heavy ion collisions; BCS pairing; BEC-BCS crossover; viscosity; imbalanced paired systems; cold atomic plasmas.

1. Introduction Over the past decade, physicists have created new states of quantum matter ranging from ultracold trapped atomic gases1 to deconfined quark-gluon plasmas produced in ultrarelativistic heavy ion collisions.2 The former, including both Bose–Einstein condensed systems and Bardeen–Cooper–Schrieffer paired fermion systems, are at characteristic temperatures of nanokelvin to millikelvin, and indeed are the coldest many-body systems in the universe. Quark-gluon plasmas on the other hand have characteristic temperatures of hundreds of Mev (∼ 1012 K), comparable to the temperature in the early universe about a microsecond after the big bang. Although these new states of matter are separated by some 21 orders of magnitude in energy scales, they have certain intriguing overlaps, on which I focus in this talk. Ultracold trapped atomic systems and the matter in the collision volume of ultrarelativistic heavy ion collisions share a number of common features. Both involve small clouds with many degrees of freedom, from of order 104 in collisions to some 106 particles in atomic clouds. Two connections between the two disparate fields are immediately apparent. First, the crossover from hadrons to quarks at finite baryon density and the crossover from Bose–Einstein condensation of molecules to BCS paired superfluids in two component cold fermionic gases bear suggestive similarities, opening fertile new approaches to the hadronization transition. Second, in expansion of the collision volume in heavy-ion collisions and in the free expansion of ultracold strongly interacting atomic Fermi systems, both systems are characterized by almost ideal hydrodynamic behavior with unusually small first viscosities. Another important connection is the physics of BCS pairing in imbalanced Fermi

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systems, where one Fermi sea is larger than the other; here recent experiments with cold atoms are providing insight for clarifying the analogous and troubled theoretical problem of BCS pairing of more massive strange with lighter up or down quarks. And in a quite different key, ultracold ionized atomic plasmas present an opportunity to carry out controlled laboratory experiments on strongly coupled plasmas, to shed light on dynamics of strongly coupled quark-gluon plasmas. Although the connections between the two systems do not appear to arise from deep underlying theoretical similarities, e.g., sharing the same universality class, study of cold atoms can inform the study of quark-gluon plasmas, and indeed vice versa.3

2. Ultracold Trapped Atom Physics Let us first recall some of the elementary physics of ultracold trapped gases. Experiments in atomic systems are done primarily on vapors of alkali metal atoms. The alkalis have one electron outside a closed shell and an odd number of protons; thus the statistics a given isotope obeys is governed by whether the nucleus has an even number of neutrons, in which case the atom is a boson, or odd, in which case it is a fermion. Since odd-odd nuclei tend to be unstable, most alkali atoms are bosons. The principal actors in the study of atomic Bose–Einstein condensates are 87 Rb (half-life, T1/2 = 4.75 × 1010 y) and 23 Na. Two fermionic alkalis are long enough lived to be easily used in trapping experiments, 6 Li, which is stable, and 40 K, with T1/2 = 1.3 × 109 y. Typical clouds contain ∼ 106 atoms, with density ∼ 1014 /cm3 , cooled via a combination of laser and then evaporative cooling, to temperatures as low as ∼ 10−9 K. The early days of cold atom experiments focussed on phenomena described accurately within mean-field theory, via the Gross–Pitaevskii equation for the condensate wave function. Such studies included the shape of the condensate (primarily described by Thomas–Fermi theory),4 the elementary modes, e.g., breathing, quadrupole oscillations,5 and shorter wavelength sound propagation. Studies of twobody correlations, through measurements of rates at which three atoms “recombine” into a molecule and a fast atom, provided direct evidence that the systems were Bose-condensed and not simply condensed in space. Noteworthy is the measurement of Ref. 6 of the single particle correlation function hψ † (~r )ψ(~r 0 )i in trapped 87 Rb, which below the transition temperature extends to a finite value at large |~r − ~r 0 |, as predicted for a Bose-condensed superfluid, but never actually observed in superfluid 4 He. The first experiment to demonstrate directly the quantum coherence of Bose condensates is that at MIT of Andrews et al.7 This experiment cooled two independent atomic systems to below the transition temperature, and then released them so that they expanded into each other, exhibiting quantum interference, analogous to the interference of two classical electromagnetic waves from independent sources. In the past several years, the field has undergone dramatic expansion, as experiment has gone into new regimes, including strong coupling and trapped BCS paired fermions. Trapped bosons and fermions behave similarly at high temperatures. At

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low temperatures, condensed bosons fall to the center of the trap, limited only by the uncertainty principle, while trapped fermionic clouds below the Fermi degeneracy temperature, Tf , are supported at larger radii by Fermi degeneracy pressure. Temperatures achieved in Fermi systems are as low as a few percent of Tf ,8 in systems of some 106 atoms. 3. Ultrarelativistic Heavy Ion Collisions As matter is heated or compressed its degrees of freedom change from composite to more fundamental. For example, by heating or compressing a gas of atoms, one eventually forms a plasma in which the nuclei become stripped of the electrons, which go into continuum states forming an electron gas. Similarly, when nuclei are squeezed, as happens in the formation of neutron stars in supernovae where the matter is compressed by gravitational collapse, the matter merges into a continuous fluid of neutrons and protons. Since nucleons themselves are made of quarks, a system of nucleons, when squeezed or heated, turns into an entirely new state of matter, a liquid of uniform quark matter — the quark-gluon plasma — composed of quarks, and at finite temperature, antiquarks and gluons as well, which are no longer confined in individual hadrons but are free to roam over the entire volume of the deconfined region. Figure 1 is an early sketch of the phase diagram of nuclear matter at finite temperature vs. baryon density,9 while Fig. 2 is a more current phase diagram for realistic up, down, and strange quark masses, in which the dashed line is a crossover and the solid line a first order transition between the confined and deconfined phases;10 the diagram also shows the region in which quarks are expected to undergo BCS pairing.11

temperature

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deconfined quarks and gluons hadrons quark superconductor

baryon chemical potential Fig. 1. Early phase diagram of equilibrated nuclear matter in the baryon density–temperature plane.

Fig. 2. Modern phase diagram using realistic quark masses, showing region of color superconductivity at high densities and low temperatures.

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Experiments at the Brookhaven Relativistic Heavy Ion Collider (RHIC) have been studying, since the year 2000, the matter produced in ultrarelativistic collisions of heavy ions (primarily 197 Au) at beam energies of 100 GeV per nucleon in the center of mass. The primary goal of the experiments is to understand how matter behaves at ultrahigh energy densities — hitherto uncharted regions of high energy density and baryon density over large spatial distances — and secondarily to produce and elucidate the quark-gluon plasma. Starting in 2008, heavy ion collisions will be studied at the LHC at CERN, at energies of 2700 TeV per nucleon in the center of mass. The projectile and target nuclei in a central ultrarelativistic collision, rather than “stopping”, pass through each other, becoming highly excited and leaving between them a state of the vacuum with extensive excitation of quark-gluon degrees of freedom. This matter is inferred to come rapidly into local thermal equilibrium as a quark-gluon plasma. The matter then cools in the post-collision expansion of the system, passing through the hadronization transition, and finally becoming free-streaming particles, primarily pions, which one detects. Since quark-gluon plasmas have a large number of helicity states (spin, color, flavor, particle-antiparticle for quarks, and eight colors for gluons), they have a much higher entropy at a given temperature than hadronic matter. Low temperature quark matter, as could possibly occur in the cores of neutron stars, would consist of degenerate Fermi seas of the lighter up and down, and heavier strange quarks. Collisions at RHIC are observed to produce matter with energy densities ∼ 5 GeV/fm3 , some 20-30 times the energy density of ordinary nuclei ∼ 0.15 GeV/fm3 (the nucleonic rest mass).12 At such energy densities the matter is certainly a quarkgluon plasma, no longer describable in terms of interacting neutrons, protons, and other laboratory hadrons. Fast quarks produced in collisions that traverse the collision volume are observed to lose energy rapidly, indicating that the matter produced is relatively opaque.13 Furthermore, pressure builds up rapidly, as indicated by large collective flow14 and fast thermalization, leading to hydrodynamic flow with very small viscosity.15

4. Common Problems of Ultracold Atom Physics and Ultrarelativistic Heavy Ion Physics Quark-gluon plasmas are always strongly interacting; by means of Feshbach resonances, atomic systems can also be made strongly interacting. The qcd running fine structure constant, αs (p) = g 2 /4π = 6π/(33 − 2Nf ) ln(p/Λ) is never small [here p is the momentum scale, Nf is the number of flavors of quarks below threshold, and Λ is the qcd scale parameter, of order 150 MeV]. Even at grand unified scales, ∼ 1015 GeV, the coupling constant g is ∼ 1/2 (compared with the charge on the electron, e ∼ 1/3, in the same units, ~ = c = 1). The interactions in cold atoms are very short ranged compared with the interatomic spacing, and because temperatures are so low, the effective atom-atom interaction can be described by a short

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range s-wave pseudopotential, V (r) = (4π~2 a/m)δ(r), where a is the s-wave atomatom scattering length. Through atomic Feshbach resonances one can, by varying the external magnetic field, dial the effective interactions over the entire range from weakly repulsive to strongly repulsive to strongly attractive to weakly attractive, thus enabling production of strongly interacting atomic systems, characterized by the scattering length a becoming large compared with the interparticle spacing. [In essence a Feshbach resonance can occur when two atoms scatter, via a hyperfine interaction, into an intermediate state with different arrangements of electronic and nuclear spins, and thus with different magnetic moments than in the initial state; by varying the magnetic field, the energies of the intermediate and initial scattering states can be made to coincide, producing a resonance.] The resonance regime is also known as the unitarity regime. Quark-gluon plasmas and qcd more generally, have non-trivial infrared physics. In qcd one can see hints of infrared trouble in the low momentum divergence of the perturbative running coupling constant, αs , above. The increase in coupling strength at low momentum scales gives rises to confinement of quarks and gluons, rendering finite order perturbation theory invalid. An analogous infrared problem, although not as severe, arises in calculating the Bose–Einstein transition temperature, T c , of a weakly interacting Bose gas, where even though the shift of Tc due to interactions is linear in the scattering length,16 a, the shift evaluated in perturbation theory diverges in the infrared in each order of perturbation theory, requiring in principle a summation of an infinite set of diagrams, and in practice numerical simulation17 as in qcd. Both strongly coupled quark-gluon plasmas and cold atom systems are effec4/3 tively scale free. The free energy density of a finite temperature plasma is ∼ n exc , where nexc is the density of excitations. Near the atomic unitarity regime, there is a clean separation of length scales; the range of the interatomic interactions, multi˚ Angstroms, is very small compared with the spacing between atoms, ∼ n1/3 , where n is the particle density, and is an irrelevant length. The scattering length, describing two particle physics at large distances, also becomes irrelevant to many-body physics when it is large compared with n1/3 . Thus the only relevant length scale at unitarity is the interparticle spacing itself. For example, the total energy per particle of a degenerate gas is E = (3Ef /5)(1 + β), where Ef = kf2 /2m is the free particle Fermi energy, kf = (3π 2 n)1/3 is the Fermi momentum, and β is a universal constant, not calculable perturbatively, of order −0.56 from Monte Carlo calculation,18 as well as diagram summation,19 expansion in the number of dimensions,20 and experiment.21 4.1. BEC-BCS crossover and the deconfinement transition Figure 3 shows the finite temperature phase diagram of a gas of two equally populated hyperfine states of ultracold fermionic atoms. The horizontal axis is −1/k f a, where a is the s-wave scattering length, and kf is the Fermi momentum of the gas.

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Free fermions +two-fermion molecules a >0 Tc } 0.23Ef BEC of di-fermion molecules

Free fermions a 0 is embedded in a homogeneous superfluid extending to infinity on both sides of the barrier. Although the profile of the barrier is one-dimensional (with the x direction orthogonal to the barrier), the slab is fully three-dimensional as it extends along the two other (y and z) directions parallel to the barrier. In addition, we regard the fermionic attraction to extend unmodified in the barrier region, a case relevant for ultracold Fermi atoms. To describe the effects of the potential barrier on the order parameter ∆(r)

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(where r = (x, y, z)), we solve the Bogoliubov–de Gennes (BdG) equations for the two-component fermionic wave functions:6 uν (r) uν (r) H(r) ∆(r) . (1) = ν vν (r) vν (r) ∆(r)∗ −H(r) Here, H(r) = −∇2 /(2m)+V (x)−µ where m and µ are the fermion mass and chemical potential. The function ∆(r) is determined via the self-consistent condition: X ∆(r) = g uν (r)vν (r)∗ (2) ν

where −g is the strength of the local fermionic attraction, eliminated eventually in favor of the scattering length aF .7 These equations have previously been considered in the weak-coupling (BCS) limit, to determine the Josephson current especially for one-dimensional situations.8 The self-consistent solution of the BdG equations (1) proceeds by extending to arbitrary values of the fermionic attraction the numerical approach introduced in Ref. 9 for weak coupling. Numerical calculations have been performed by imposing either the value n0 q/m of the Josephson current (provided it can be self-consistently sustained) where n0 is the bulk density, or the value of the asymptotic phase difference δφ = φ(x = +∞) − φ(x = −∞) between the two sides of the barrier. Under these circumstances, we write ∆(x) = |∆(x)| exp[2iqx+iφ(x)]. Figures 1(a) and 1(b) present the spatial variation of the magnitude |∆(x)| and the phase φ(x) of the order parameter across a barrier centered at x = 0, for three characteristic values of (kF aF )−1 and for the corresponding maximum value of the Josephson current. Note the depression of |∆(x)| from its bulk value |∆0 | due to the presence of the repulsive barrier, which is accompanied by a sharp variation of the phase φ(x) across the barrier in order to keep the current uniform, thus resulting in the total phase difference δφ accumulating across the barrier. Both the depression of |∆(x)| at x = 0 and the value of δφ systematically increase from weak to strong coupling. These features are reflected in the corresponding density profiles n(x) shown in Fig. 1(c), a quantity which is directly measurable with ultracold Fermi 0.4 1

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atoms. In the BCS limit, Friedel oscillations modulated by 2kF affect n(x) as well as |∆(x)| and φ(x). The characteristic Josephson current/phase relation J(δφ) for different couplings encompassing unitarity is shown in Fig. 2(a), for the same values of L and V0 considered in Fig. 1. Note that the Josephson current is considerably enhanced at unitarity with respect to the BCS and BEC sides, and that the curves J(δφ) stretch from being proportional to sin(δφ) when (kF aF )−1 = 1.0 to being (almost) proportional to cos(δφ/2) when (kF aF )−1 = −1.0. The contribution of the discrete bound levels, known as Andreev–Saint James states,10 to the maximum Josephson current is shown in Fig. 2(b) as a function of the x coordinate. As one can see, this contribution (localized near the barrier) is most relevant at unitarity where the total current is also largest. In the light of these preliminary results, the evolution of the Andreev–Saint James states through the BCS–BEC crossover deserves a more detailed study in the future. As for any approach to a crossover problem, the present study of the Josephson effect has also to rely on a definite benchmark in the strong-coupling (BEC) limit. In this context, it has been shown11 that the fermionic BdG equations (1) can be suitably mapped onto the Gross–Pitaevskii (GP) equation.12,13 In Fig. 2(c) we show a comparison of the Josephson characteristic obtained from the BdG equations with the indipendent solution of the GP equation for the same geometry. From these Josephson characteristics one can extract the maximum (critical) current Jc = n0 qc /m associated with the given barrier and coupling. The corresponding critical velocity qc /m is shown in Fig. 3 vs (kF aF )−1 from weak to strong coupling and for several barrier heights. All curves present a maximum at about unitarity, with the value of the maximum progressively increasing as the ratio V0 /EF is decreased. As shown in Fig. 3, in the limit of vanishing small barriers height the maximum velocity reaches the Landau critical velocity which is, in turn, determined by the available quasi-particle excitations that are of a different nature on the two sides of the crossover. Specifically, on the BCS side (“left” Landau branch) there occur the

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p pair-breaking excitations characteristic of BCS theory, yielding qc2 /m = µ2 + ∆20 − µ; in the BEC side one expects sound-mode quanta to be the relevant excitations (the “right” Landau branch in Fig. 3 corresponds to the sound velocity of the Bogoliubov–Anderson mode obtained within the BCS-RPA approximation,14 which generalizes the result qc2 /mB = µB valid in the BEC limit).

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Finally, we point out that by extending to the whole BCS–BEC crossover the experimental procedure followed in Ref. 15, one may test our prediction that the superfluid flow is intrinsically more robust at unitarity, a regime which should therefore be preferred for applications based on fermionic superfluid flow. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

C. Chin et al., Science 305, 1128 (2004). M. Greiner, C.A. Regal, and D.S. Jin, Phys. Rev. Lett. 94, 070403 (2005). M.W. Zwierlein et al., Nature 435, 1047 (2005). C.H. Schunck et al., cond-mat/0702066. A. Spuntarelli, P. Pieri, and G.C. Strinati, arXiv:0705.2658v2[cond-mat.other]. P.G. De Gennes, Superconductivity of Metals and Alloys (Benjamin, New York, 1966). C.A.R. S` a de Melo, M. Randeria, and J.R. Engelbrecht, Phys. Rev. Lett. 71, 3202 (1993). A. Furusaki and M. Tsukada, Phys. Rev. B 43, 10164 (1991). See also G. Wendin and V.S. Shumeiko, Phys. Rev. B 53, R6006 (1996), and references quoted therein. R.A. Riedel, Li-Fu Chang, and P.F. Bagwell, Phys. Rev. B 54, 16082 (1996). A. Andreev, Soviet Phys. JETP 19, 1228 (1964); D. Saint-James, J. de Physique 25, 899 (1964). P. Pieri and G.C. Strinati, Phys. Rev. Lett. 91, 030401 (2003). E.P. Gross, Il Nuovo Cimento 20, 454 (1961); J. Math. Phys. 4, 195 (1963). L.P. Pitaevskii, Sov. Phys. JETP 13, 451 (1961). M. Marini, F. Pistolesi, and G.C. Strinati, Eur. Phys. J. B 1, 151 (1998). P. Engels and C. Atherton, arXiv:0704.2427v1[cond-mat.other].

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ULTRA-COLD DIPOLAR GASES CHIARA MENOTTI ICFO – Institut de Ci` encies Fot` oniques, E-08860 Castelldefels, Barcelona, Spain CNR-INFM-BEC and Dipartimento di Fisica, Universit` a di Trento, I-38050 Povo, Italy MACIEJ LEWENSTEIN ICFO – Institut de Ci` encies Fot` oniques, E-08860 Castelldefels, Barcelona, Spain ICREA – Instituci´ o Catalana de Recerca i Estudis Avan¸cats, E-08010 Barcelona, Spain We present a concise review of the physics of ultra-cold dipolar gases, based mainly on the theoretical developments in our own group. First, we discuss shortly weakly interacting ultra-cold trapped dipolar gases. Dipolar Bose–Einstein condensates exhibit non-standard instabilities and the physics of both Bose and Fermi dipolar gases depends on the trap geometry. We focus then the second part of the paper on strongly correlated dipolar gases and discuss ultra-cold dipolar gases in optical lattices. Such gases exhibit a spectacular richness of quantum phases and metastable states, which may perhaps be used as quantum memories. We comment shortly on the possibility of superchemistry aiming at the creation of dipolar heteronuclear molecules in lattices. Finally, we turn to ultra-cold dipolar gases in artificial magnetic fields, and consider rotating dipolar gases, that provide in our opinion the best option towards the realization of the fractional quantum Hall effect and quantum Wigner crystals. Keywords: Ultra-cold dipolar gases; metastable states; rotating dipolar gases; fractional quantum Hall effect (FQHE).

1. Introduction This paper has been presented as an invited lecture at the International Conference on Recent Progress on Many Body Theories, RPMBT 2007, held in Barcelona in July 2007, in the session devoted to ultra-cold atoms. This conference puts traditionally a lot of emphasis on the development of new methodologies, analytic and numerical methods for many body problems. The lecture given by M. Lewenstein was a little different in character: instead of talking about some new specific method, M. Lewenstein gave a broad review of the topic, trying to convince the audience that ultra-cold dipolar gases provide a fantastic playground to apply modern many body theory. This topic belongs to one of the hottest areas of modern atomic, molecular, and optical (AMO) physics, and giving a full review in a one hour lecture is impossible. M. Lewenstein based his presentation mainly on the activities of his own group,

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which fortunately touch most of the aspects of the physics of ultra-cold dipolar gases (UDG), mentioning only some selected important contributions from other groups. In this sense the present paper is not a review in the strict sense; it is more like a review of important subtopics within the main topic. Outline. The outline of this paper is thus the following. It consists of three, relatively independent parts. In the first part we introduce UDGs, and argue why they are so interesting and how to realize them in the laboratory. We sketch very briefly the physics of weakly interacting trapped dipolar Bose and Fermi gases, and talk about the influence of the trap geometry on the physical properties of the UDGs. The second subject of the paper concerns ultra-cold dipolar gases in optical lattices, that are examples of strongly correlated systems. Such gases exhibit a spectacular richness of quantum phases (Mott insulators and insulating checkerboard phase, superfluid and supersolid phases), as well as an extravagantly large variety of metastable states, which may perhaps be used as quantum memories. We comment here shortly on the possibility of superchemistry aiming at the creation of dipolar heteronuclear molecules in lattices. In the third and last part, we turn to ultra-cold dipolar gases in artificial magnetic fields and consider rotating dipolar gases, that provide in our opinion the best option towards the realization of the fractional quantum Hall effect and quantum Wigner crystals.

2. Weakly Interacting Trapped Dipolar Gases Why dipolar gases? Some of the most fascinating experimental and theoretical challenges of modern atomic and molecular physics arguably concern ultra-cold dipolar quantum gases.1 The recent experimental realization of a quantum degenerate dipolar Bose gas of Chromium,2 and the progress in trapping and cooling of dipolar molecules3 have opened the path towards ultra-cold quantum gases with dominant dipole interactions. In particular just before the Barcelona RPMBT conference, the group of Tilman Pfau in Stuttgart realized a UDG of Chromium with dominant magnetic dipole interactions, employing a Feshbach resonance to “turn off” the short range Van der Waals forces.4 Several groups have reported in 2007 enormous progresses in trapping and manipulating mixtures of different atomic species in an optical lattice Ref. 5. Such systems realize the first step towards the superchemistry of UDGs, that we discuss in the second part of this paper. Why are dipole interactions interesting? Because of their very nature. For most of the systems studied so far, one assumes the dipole moment to be polarized, i.e. oriented in the same direction via applying either magnetic or electric fields. In such case the dipolar potential between two particles is

V (r) =

d2 1 − 3 cos2 θ , 3 r

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Fig. 1. Schematic representation of the anisotropic character of dipole-dipole interaction between dipoles oriented vertically (red arrows): for relative distances along the orientation of the dipoles the interaction is attractive (green arrows), while for relative distances perpendicular to the orientation of the dipoles the interaction is repulsive (blue arrows).

where r is the inter-particle distance, θ is the angle between the direction of the dipole moment and the vector connecting the particles. The interaction is anisotropic and partially attractive. In particular, if the two dipoles are on top of one another, they attract themselves, if they are aside, they repel each other (see Fig. 1). Locating the dipoles in a vertical cigar-shaped trap leads to a mainly attractive gas that should therefore exhibit collapse. Locating the dipoles in a horizontal pancake trap leads to a mainly repulsive gas and should allow to at least partially stabilise the system. Several groups have attacked the theory of the UDGs starting from 1999 (for reviews see1,6 ). Particularly important were the pioneering papers by G´ oral, Pfau and Rz¸az˙ ewski,7 the L. You group Ref. 8, or the G. Kurizki group.9 Trapped dipolar Bose gases. The dependence of the physics of trapped dipolar gases on geometry is quite strong and leads to dramatic effects,10 as described in the following and summarised in Fig. 2. Let us for simplicity consider a dipolar gas of N Bose particles trapped in a harmonic trap, with all of the dipoles d oriented in the same direction, and assume that the particles interact dominantly via dipoledipole interaction. This means, for instance, that we are dealing with heteronuclear molecules with sufficiently large electric dipoles, or a Chromium gas with small magnetic moments interactions, but even smaller s-wave scattering length. At sufficiently low temperatures such a gas will undergo condensation, and its behaviour is well described by the Gross–Pitaevskii equation (this fact is by no means obvious). Let us first consider cigar-shaped (ωρ ≥ ωz ) and ”soft” pancake-shaped traps with (ωρ ≤ ωz ). There appears a kind of quantum phase transition as a function of the trap aspect ratio λ∗ = (ωρ /ωz )1/2 ' 0.4, above which the sign of the energy of the mean dipole interaction V changes from positive to increasingly negative, as we increase N d2 for λ∗ < λ ≤ 1, and remains always increasingly negative for λ > 1.

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The condensate becomes more and more cigar-shaped, until it undergoes a collapse, somewhat similar to what is occurring for a gas with negative scattering length, i.e. effectively attractive Van der Waals interactions.11 For hard pancake traps with λ < λ∗ , V grows as we increase N d2 and the gas is dominantly repulsive. There is no standard collapse and BEC with much larger values of N are stable. The condensate aspect ratio decreases with N d 2 . For ωρ ωz , one can distinguish two regimes: i) for ωρ V ωz , we deal with a quasi-2D Bose gas with repulsive interactions, which attains radially a parabolic Thomas–Fermi profile; ii) for V ≥ ~ωz , we deal with the 3D gas in the Thomas–Fermi regime. Here the gas does feel the attractive part of the dipolar interactions and undergoes a short wavelength instability, which leads to a roton-maxon minimum and then instability in the excitations spectrum12 (see Fig. 3).

Trapped dipolar Fermi gases. The dependence of the physics of trapped dipolar gases on the geometry leads also to a quantum phase transition in the case of Fermi gases.13 Let us consider again for simplicity a dipolar gas of N Fermi particles trapped in a harmonic trap, with all of the dipoles d oriented in the same direction, and interactions being dominantly of dipole-dipole kind. The question is whether at sufficiently low temperatures such a gas will undergo a transition to a superfluid state (Bardeen–Cooper–Schrieffer (BCS) state), and whether its behaviour is well described by the BCS equations (again, this latter fact is by no means obvious). Pioneering papers on this subject were written by the groups of L. You and H. Stoof, who have looked at the possibility of p-wave pairing,14 and the group of K. Rz¸az˙ ewski, who studied the Thomas–Fermi theory.15 The BCS theory in homogeneous dipolar gas was investigated in detail by Baranov and Shlyapnikov.16 In Ref. 13, we have looked at the BCS transition in a trap, and have shown

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indeed the existence of a critical aspect ratio, similarly as in the case of a Bose gas. The phase diagram is presented in Fig. 4 as a function of λ−1 and dipole interactions in units of Fermi energy Γ. It can be viewed in two ways: for a given λ the systems undergoes a transition from the normal to the superfluid state as the dipole interactions grow. Conversely, for a fixed dipole interactions the system undergoes the normal-superfluid transition as λ decreases. For very small dipole interactions this transitions occurs in a region of parameters that goes beyond the applicability of our theory.

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3. Dipolar Bose Gases in Optical Lattices Ultra-cold gases in optical lattices. Ultra-cold atomic gases in optical lattices (OL) are nowadays the subject of very intensive studies, since they provide an unprecedented and unique possibility to study numerous challenges of quantum many body physics (for reviews see17,18 ). In particular, such systems allow to realize various versions of Hubbard models,19 a paradigmatic example of which is the Bose– Hubbard model.20 This model exhibits a superfluid (SF) — Mott insulator (MI) quantum phase transition,21 and its atomic realization has been proposed in the seminal paper of Ref. 22, followed by the seminal experiments of Ref. 23. Several aspects and modifications of the SF-MI quantum phase transition, or better to say crossover,24 have been intensively studied recently (cf.18,25 ). Ultra-cold dipolar gas in a lattice. We have proposed to look at the UDG in a 2D lattice in 2002.26 The Hamiltonian of the system differs from the standard Hubbard-Bose model described by the Hamiltonian H = −t

X † UX ni (ni − 1) − µN, [bi bi−1 + h.c.] + 2 i i

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P P where N = i ni = i b†i bi is the atom number operator, t is the hopping term, and U denotes the strength of on-site interactions. A UDG in a lattice is described by an extended Bose–Hubbard Hamiltonian H = −t

X † UX U1 X U2 X ni (ni −1)+ ni nj + ni nj +...−µN, [bi bi−1 +h.c.]+ 2 i 2 2 i

(3) where the sum over hi, ji pertains to nearest neighbours, the one over hhi, jii to nextnearest neighbours, etc., and Ui are determined by dipole-dipole interactions. To a good approximation, assuming that all dipoles are perpendicular to the plane of the 3 lattice, Un = d2 /rij , where rij are the distances between the involved sites; generally it is given by the expression in Eq. (1). The resulting model exhibits a rich variety of quantum phases: apart from the standard superfluid and Mott insulator states, it can form a checkerboard phase at half filling, or close to it in the parameter space a supersolid state, i.e. superfluid with a periodic density modulation in the density and in the order parameter. It can also form a collapsing state if the interactions are too attractive.26 The possibility of realizing a supersolid state with ultra-cold atoms is at present particularly attractive because, to our knowledge, its existence, claimed in 4 He experiments,27 is still controversial.28 Experiments with ultra-cold dipolar atoms might provide a much cleaner environment for the creation and observation of such phases. Metastable states. Most recently, we have pointed out for the first time that a lattice system with long-range interactions presents many insulating metastable

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states in the low tunnelling part of the phase diagram.29 The metastable states arise as local minima of the energy. We access them using a mean-field approach and a time dependent Gutzwiller ansatz, which allows to study the dynamics of the system both in real and imaginary time. The imaginary time evolution, which mimics dissipation in the system, converges unambiguously to the ground state of the system for the Bose–Hubbard model in presence of on-site interaction only. In the presence of long-range interaction it shows a strikingly different behaviour and converges often to different configurations, depending on the exact initial conditions. In this way, we clearly get a feeling of the existence of metastable states in the system. In the real time evolution, their stability is confirmed by typical small oscillations around a local minimum of the energy. For all the insulating metastable configurations, we calculate the insulating lobes in the J − µ phase space, using a mean-field perturbative approach. This results in a much more complex phase diagram, as shown in Fig. 5. The metastable states have a finite lifetime due to the tunnelling to different metastable states. We have used a path integral approach in imaginary time, combined with a dynamical variational method to estimate this lifetime, which results to be very long for small tunnelling parameter J and large systems. However, for large systems the number of the metastable states and the variety of their patterns is so large, and their energy separation so small, that it turns out to be very difficult to control the presence of defects. We have checked that by using superlattices one can prepare the atoms in configurations of preferential symmetry with very small uncertainty. If a given configuration corresponds to a metastable state, it survives also once the superlattice is removed, due to dipoledipole interaction. For the detection scheme we have presently in mind, it is also essential (not in line of principle, but practically for present experimental possibilities) to create a given configuration in a reproducible way. In fact, the spatially modulated structures characterising the metastable states can be detected via the measurement of the noise correlations of the expansion pictures,30–32 which equal the modulus square of the Fourier transform of the density distribution in the lattice and is in principle able to recognise the periodic modulations or the defects in the density distribution. However, since the signal to noise ratio required for single defect recognition is beyond the present experimental possibilities, one should average over a finite number of different experimental runs producing the same spatial distribution of atoms in the lattice, and hence accurate reproducibility is required. In Fig. 6, we show the noise correlations for the metastable configurations at filling factor 1/2 shown in Fig. 5, (I) to (III). Presently we are studying the possibility of transferring in a controlled way those systems from one configuration to another. This, together with the capability of initialising and reading out the state of the lattice, may make those systems useful for applications as quantum memories.

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“Superchemistry” of dipolar heteronuclear molecules in an optical lattice. The idea is to create heteronuclear molecules starting from a Mott insulating phase with exactly one atom per species per site.33 Polar dimers would be then formed by photo-association or by using a Feshbach resonance. With an appropriate choice of of scattering lengths, such that the two species do not remain immiscible, one can demonstrate that the two-species Mott state is the ground state of the system and can be reached starting by a two-component superfluid and slowly ramping up the optical lattice potential (see Fig. 7).

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Fig. 8. Quantum melting of 41 K −87 Rb dimers initially in the MI phase towards the ultracold dipolar molecular BEC; the plot shows the time evolution of the molecular superfluid order parameter |hbi i| (solid line), which is the same for all lattice sites. The dashed line refers to static calculations of the ground state of the dipolar molecules placed in the lattice.

Before the creation of molecules, the system is described by a two-species Bose– Hubbard Hamiltonian with local interactions i X Xh nai nbi H= Ja a†i aj + Jb b†i bj + Uab hi,ji

i

1X + [U0a nai (nai − 1) + U0b nbi (nbi − 1)] , 2 i

(4)

with a and b denoting the two species, while, after the creation of the molecules, assuming molecules with non negligible dipole moment, the Hamiltonian is the one of a single molecular component gas with long-range interactions, as written in Eq. (3). Finally, this Mott insulating state of dipolar molecules can be melted to a superfluid heteronuclear molecular condensate, as shown in Fig. 8.

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4. Ultra-cold Gases in Artificial Gauge Fields Rotating ultra-cold gases. It is well known that rapidly rotating harmonically trapped gases of neutral atoms exhibit effects completely analogous to charged particles in uniform magnetic fields (for a recent overview, see for instance34 ). In particular, one should be able to realize analogues of the fractional quantum Hall effect (FQHE) in such systems.35,36 Particularly interesting in this context are rotating dipolar gases (RDG). Bose–Einstein condensates of RDGs exhibit novel forms of vortex lattices, e.g., square, stripe- and bubble-“crystal” lattices.37 The stability of these phases in the lowest Landau level was recently investigated .38 We have demonstrated that the quasi-hole gap survives the large N limit for fermionic RDGs.39 This property makes them perfect candidates to approach the strongly correlated regime, and to realize Laughlin liquids Ref. 40 at filling ν = 1/3, and quantum Wigner crystals at ν ≤ 1/741 for a mesoscopic number of atoms N ' 50 − 100. Lately, Rezayi et al.42 have shown that the presence of a small amount of dipoledipole interactions stabilizes the so-called bosonic Rezayi-Read state at ν = 3/2 whose excitations are both fractional and non-Abelian.

Ordered structures in rotating Bose gases. In the recent two years our group has concentrated on studies of small samples of rotating atoms using exact diagonalization methods. These early studies dealt with the description of ordered structures in rotating ultra-cold Bose gases (by looking at the single particle density matrix and pair correlation function34 ), and by studying symmetry breaking in small rotating clouds of trapped ultra-cold Bose atoms.43 The characterization of small samples of cold bosonic atoms in rotating micro-traps has recently attracted increasing interest due to the possibility to deal with a few number of particles per site in optical lattices. In the Ref. 34 we considered two-dimensional systems of few cold Bose atoms confined in a harmonic trap in the XY plane, and submitted to strong rotation around the Z axis. By means of exact diagonalization, we analysed the evolution of the ground state structures as the rotational frequency Ω increases. Various kinds of ordered structures were observed. In some cases, hidden interference patterns exhibit themselves only in the pair correlation function; in some other cases explicit broken-symmetry structures appear that modulate the density. The standard scenario, valid for large systems (i.e., nucleation of vortices into an Abrikosov lattice, melting of the lattice, and subsequent appearance of fractional quantum Hall type states up to the Laughlin state), is absent for small systems of N < 10 atoms, and only gradually recovered as N increases. On the one hand, the Laughlin state in the strong rotational regime contains ordered structures much more similar to a Wigner crystal or a molecule than to a fermionic quantum liquid. This result has some similarities to electronic systems, extensively analysed previously. On the other hand, in the weak rotational regime, the possibility to obtain equilibrium states whose density reveals an array of vortices is restricted to some critical values of the rotation frequency Ω.

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Rotational symmetry breaking. In Ref. 43 we have studied the signatures of rotational and phase symmetry breaking in small rotating clouds of trapped ultracold Bose atoms by looking at the rigorously defined condensate wave function. Rotational symmetry breaking occurs in narrow frequency windows, where energy degeneracy between the lowest energy states of different total angular momentum takes place, and leads to a complex condensate wave function that exhibits vortices clearly seen as holes in the density, as well as characteristic vorticities. Phase symmetry (or gauge symmetry) breaking, on the other hand, is clearly manifested in the interference of two independent rotating clouds. Ultra-cold rotating dipolar Fermi gases. Armed by the experience on rotating Bose gases with short range interactions, we considered in the recent Letter 44 a system of N dipolar fermions rotating in an axially symmetric harmonic trapping potential strongly confined in the direction of the axis of rotation. Along this z-axis, the dipole moments, as well as spins are assumed to be aligned. Various ways of experimental realization of ultra-cold dipolar gases are discussed in.1 In case of low temperature T and weak chemical potential µ with respect to the axial confinement ωz , the gas is effectively 2D, and the Hamiltonian of the system in the rotating reference frame reads

H=

N X M 2 1 (~ pj −M Ω~ez × ~rj )2 + ω0 − Ω2 rj2 + Vd . 2M 2 j=1

(5)

Here, ω0 ωz is the radial trap frequency, Ω is the frequency of rotation, M is PN d2 the mass of the particles, Vd = j 0). 2 These are the largest diatomic molecules obtained so far, with the size of thousands of angstroms. Being highly excited, they are remarkably stable with respect to collisional relaxation into deep bound states, which is a consequence of the Pauli exclusion principle for identical fermionic atoms.3 For example, at densities of about 1013 cm−3 the relaxation time can exceed seconds, which is more than four orders of magnitude longer than the life time of similar molecules consisting of bosonic atoms. Until now, experiments have been done either with 6 Li or with 40 K atoms in two different hyperfine states. Currently, a new generation of experiments is being set up for studying mixtures of different fermionic atoms, with the idea of revealing

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the influence of the mass difference on superfluid properties and finding novel types of superfluid pairing. Weakly bound heteronuclear molecules on the positive side of the resonance are unique objects,4,5 which should manifest collisional stability and can pave a way to creating ultracold dipolar gases. So far it was believed that dilute Fermi mixtures should be in the gas phase, like Fermi gases of atoms in two different internal states. In this paper we find that the system of molecules of heavy (mass M ) and light (mass m) fermions can undergo a phase transition to a crystalline phase. This is due to a repulsive intermolecular potential originating from the exchange of light fermions and inversely proportional to m. As the kinetic energy of the molecules has a prefactor 1/M , above a certain mass ratio M/m the system can crystallize. We show that the interaction potential in a sufficiently dilute system of molecules is equal to the sum of their pair interactions, which we derive using the Born– Oppenheimer approximation. All further analysis is performed for the case where the motion of heavy atoms is confined to two dimensions, whereas the light fermions can be either 2D or 3D.6 We calculate the zero-temperature gas-crystal transition line using Diffusion Monte Carlo (DMC) method and draw the phase diagram in terms of the mass ratio and density. This phase transition resembles the one in the model of flux lattice melting in strongly type-II materials, where the flux lines are mapped onto a system of bosons interacting via a 2D Yukawa potential.7 In this case the Monte Carlo studies8,9 identified the first order liquid-crystal transition at zero and finite temperatures. Aside from the difference in the interaction potentials, a distinguished feature of our system is related to its stability. The molecules can undergo collisional relaxation into deep bound states, or form weakly bound trimers. We analyze resulting limitations on the lifetime of the system. 2. Effective Interaction in a Many-Body System of Molecules We first derive the Born–Oppenheimer interaction potential in the system of N molecules. In this approach the state of light atoms adiabatically adjusts itself to the set of heavy-atom coordinates {R} = {R1 , ..., RN } and one calculates the wavefunction and energy of light fermions in the field of fixed heavy atoms. Omitting the interaction between light (identical) fermions, it is sufficient to find N lowest single-particle eigenstates, and the sum of their energies will give the interaction potential for the molecules. For the interaction between light and heavy atoms we use the Bethe-Peierls approach10 assuming that the motion of light atoms is free everywhere except for their vanishing distances from heavy atoms. The wavefunction of a single light atom then reads: Ψ({R}, r) =

N X i=1

Ci Gκ (r − Ri ),

(1)

where r is its coordinate, and the Green function Gκ satisfies the equation (−∇2r + κ2 )Gκ (r) = δ(r). The energy of the state (1) equals = −~2 κ2 /2m, and here we

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only search for negative single-particle energies (see below). The dependence of the coefficients Ci and κ on {R} is obtained using the Bethe-Peierls boundary condition: Ψ({R}, r) ∝ Gκ0 (r − Ri );

r → Ri .

(2)

Up to a normalization constant, Gκ0 is the wavefunction of a bound state of a single molecule with energy 0 = −~2 κ20 /2m and molecular size κ−1 0 . From Eqs. (1) and (2) P one gets a set of N equations: j Aij Cj = 0, where Aij = λ(κ)δij +Gκ (Rij )(1−δij ), Rij = |Ri − Rj |, and λ(κ) = limr→0 [Gκ (r) − Gκ0 (r)]. The single-particle energy levels are determined by the equation det [Aij (κ, {R})] = 0.

(3)

For Rij → ∞, Eq. (3) gives an N-fold degenerate ground state with κ = κ0 . At finite large Rij , the levels split into a narrow band. Given a small parameter 0 ˜ ξ = Gκ0 (R)/κ 0 |λκ (κ0 )| 1,

(4)

Aij = λδij +[Gκ0 (Rij )+κ0λ λ∂Gκ0 (Rij )/∂κ](1−δij ),

(5)

˜ is a characteristic distance at which heavy atoms can approach each other, where R the bandwidth is ∆ ≈ 4|0 |ξ |0 |. It is important for the adiabatic approximation that all lowest N eigenstates have negative energies and are separated from the continuum by a gap ∼ |0 |. We now calculate the single-particle energies up to second order in ξ. To this order we write κ(λ) ≈ κ0 + κ0λ λ + κ00λλ λ2 /2 and turn from Aij (κ) to Aij (λ): where all derivatives with respect to λ are taken at λ = 0. Using Aij (5) in Eq. (3) gives a polynomial of degree N in λ. Its roots λi give the light-atom energy spectrum i = −~2 κ2 (λi )/2m. The total energy is then given by N N N i X X X ~2 h 2 N κ0 + 2κ0 κ0λ λi + (κκ0λ )0λ λ2i . E= i = 2m i=1 i=1 i=1

(6)

The summation in Eq. (6) can be done straightforwardly keeping only the terms up to second order in ξ and using basic properties of determinants and polynomial P roots. The first order terms vanish, and finally we get E = N 0 +(1/2) i6=j U (Rij ), where i ∂G2κ0 (R) ~2 h κ0 (κ0λ )2 (7) U (R) = − + (κκ0λ )0λ G2κ0 (R) . m ∂κ Thus, up to second order in ξ the interaction in the system of N molecules is the sum of binary potentials (7). “Three-body” terms proportional, for example, to Gκ0 (Rij )Gκ0 (Rjk ) are absent. If the motion of light atoms is 3D, the Green function is Gκ (R) = (1/4πR) exp(−κR), and λ(κ) = (κ0 − κ)/4π, with the molecular size κ−1 equal 0 to the 3D scattering length a. Equation (7) then gives a repulsive potential U3D (R) = 4|0 |(1 − (2κ0 R)−1 ) exp(−2κ0 R)/κ0 R,

(8)

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and the criterion (4) reads (1/κ0 R) exp(−κ0 R) 1. For the 2D motion of light atoms we have Gκ (R) = (1/2π)K0 (κR) and λ(κ) = −(1/2π) ln(κ/κ0 ), where K0 is the decaying Bessel function, and κ−1 follows from.6 This leads to a repulsive 0 intermolecular potential: U2D (R) = 4|0 |[κ0 RK0 (κ0 R)K1 (κ0 R) − K02 (κ0 R)],

(9)

with the validity criterion K0 (κ0 R) 1. In both cases, which we denote 2 × 3 and 2 × 2 for brevity, the validity criteria are well satisfied already for κ0 R ≈ 2. 3. Phase Diagram The Hamiltonian of the many-body system reads: ~2 X 1X H =− ∆R i + U (Rij), 2M i 2

(10)

i6=j

and the state of the system is determined by two parameters: the mass ratio M/m and the rescaled 2D density nκ−2 0 . At a large M/m, the potential repulsion dominates over the kinetic energy and one expects a crystalline ground state. For separations Rij < κ−1 the adiabatic approximation breaks down. However, 0 the interaction potential U (R) is strongly repulsive at large distances. Hence, ¯ close to 2/κ0 , they apeven for an average separation between heavy atoms, R, −1 proach each other at distances smaller than κ0 with a small tunneling probability p P ∝ exp(−β M/m) 1, where β ∼ 1. We extended U (R) to R . κ−1 0 in a way providing a proper molecule-molecule scattering phase shift in vacuum and checked that the phase diagram for the many-body system is not sensitive to the choice of this extension. In order to study the phase diagram of the system we use the diffusion Monte Carlo (DMC) method, which gives an exact solution (apart from statistical uncertainty) for the many-boson ground state.11 We consider a system of N molecules described by the Hamiltonian (10) in a box of size Lx × Ly with periodic boundary conditions and density n = N/(Lx Ly ). Importance sampling is provided by the trial wavefunction in the Nosanow-Jastrow form ψT (R1 , ..., RN ) =

N Y

i=1

(0)

g1 (|Ri − Ri |)

N Y

g2 (Rjk ).

(11)

j L/2 where L = min{Lx , Ly }. Chosen in this way, the function g2 (R) and its derivative g20 (R) are continuous. The free variational parameters Apar and Bpar are defined by minimizing the variational energy. In the 2 × 2 case where the potential is given by Eq. (9), the two-body Jastrow term is chosen as o n  2 2(L−2Rpar )(L−Rpar ) R  , R ≤ Rpar exp −  2 2 L  Rparn o 2 2R (L−R ) par 1 4 1 (13) g2 (R) = exp − par , Rpar < R ≤ L/2 L(L−2Rpar ) R + L−R − L    1, R > L/2 . The long-range correlations arise from phonon excitations and are intrinsically many-body. It can be shown13 that in two-dimensional geometry this implies g2 (R) ∝ exp{−const/R} large distance asymptotic behavior. Corresponding functional dependence of g2 (R) has been used for R > Rpar in (13). Both two-body Jastrow terms, (12) and (13), have zero derivative at L/2 according to the periodic boundary conditions applied to the simulation box. The trial wavefunctions (11) calculated for many different values of the mass ratio and density of molecules are then used as guiding wavefunctions for the DMC method, which provides the exact energies of the solid and gas phases. The resulting phase diagram for N = 30 molecules is displayed in Fig. 1. We also perform the same analysis for larger N in the high density part of the phase diagram in order to ensure that finite size effects can be neglected. In the low density region of the phase diagram the de Broglie wavelength of molecules is much larger than κ−1 0 , and U (R) can be approximated by a hard-disk potential with the diameter equal to the 2D scattering length.14 The critical density for the gas-crystal phase transition in a system of 2D hard-disk bosons has been found by Lei Xing.16 Using this result and calculating the 2D scattering length as a function of M/m for the two cases (2×3 and 2×2), we approximate the phase boundary by solid curves in Fig. 1. The meaning of dashed curves is explained in Sec. 4 where we construct the harmonic approximation for the crystal.

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Fig. 1. DMC gas-crystal transition lines for 3D (triangles) and 2D (circles) motion of light atoms. Solid curves show the low-density hard-disk limit, and dashed curves the results of the harmonic approach (see Sec. 4).

4. Crystal in the Harmonic Approximation Let us denote the displacements of molecules from their equilibrium positions in (0) the crystal by ui = Ri − Ri . The harmonic approximation can be constructed if the corresponding rms size u ¯ is smaller than the characteristic lengthscale of the (0) −1 intermolecular potential κ0 . Then the quantities U (Rij ) = U (|Rij + ui − uj |) in Eq. (10) are expanded up to second order terms in the displacements ui and uj (small parameter is κ0 u ¯) leading to the harmonic Hamiltonian ~2 X 1X (0) H2 = − ~ui · Λ(Rij )uj , (14) ∆u i + 2M i 2 i6=j

where Λ is a 2 × 2 matrix and we omit terms independent of the displacements. The eigenstates of the Hamiltonian (14) are phonons, and their wavefunctions ˜ ~k) = P Λ(R(0) )e−i~k·R~ (0) i . are obtained by diagonalizing the dynamical matrix15 Λ( i i 2 Its eigenvalues M ωk,s and eigenvectors ~ek,s define the phonon energies and polarizations respectively. The wavevector ~k belongs to the Brillouin zone of the lattice and the polarization index s distinguishes the two phonon polarizations (longitudinal and transversal). Introducing the phonon annihilation and creation operators15 ak,s and a†k,s satisfying independent bosonic commutation relations the HamiltoP nian (14) can be written in the diagonal form: H2 = k,s ~ωk,s (a†k,s ak,s + 1/2). The displacement operator of molecule i is expressed in terms of aκ and a†κ by the formula q 1 X ~ ~ (0) ~ ~ (0) ~/M ωk,s ak,s~ek,s eik·Ri + a†k,s~e∗k,s e−ik·Ri . (15) ui = √ 2N k,s

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In the ground state the phonon modes are not occupied (ha†k,s ak,s i = 0) and the mean square displacement of molecules, which characterizes their zero-point motion, is given by 1 X ~ u ¯2 = . (16) 2N M ωk,s k,s

We note that the dynamical matrix Λ ∝ κ20 U is independent of M and the same 2 holds for its eigenvalues M ωk,s . Therefore, ωk,s ∝ M −1/2 and according to Eq. (16) u ¯ ∝ M −1/4 .

(17)

From Eq. (17) one concludes that within the harmonic approximation the Linde¯ ∝ (M/m)−1/4 decreases as we go up in the phase diagram in mann ratio γ = u ¯ /R Fig. 1. Obviously, for sufficiently large mass ratios we enter the regime where the harmonic approximation is valid. Quantitatively, we compare the values of γ deduced from Eq. (16) with the results obtained by the DMC method and find a good agreement between these quantities in the solid phase in the right (large density) half of the phase diagram. We also notice that in both 2×3 and 2×2 cases the parameter γ on the transition line fluctuates only slightly ranging from 0.23 to 0.27. This gives us the idea to fit the phase boundary by the mass ratio dependence on density in the harmonic approximation assuming a constant Lindemann ratio. Dashed lines in Fig. 1 show the result of this fit, and we see a reasonable agreement between the DMC and “harmonic” methods. In fact, we believe that such estimates of the position of the transition line can be very useful for generic smooth interaction potentials in various dimensions. At least, the harmonic approach can provide the first guess and thus facilitate the DMC scheme. 5. Realization and Lifetime of the Crystalline Phase The mass ratio above 100, required for the observation of the crystalline order (see Fig. 1), can be achieved in an optical lattice with a small filling factor for heavy atoms. Their effective mass in the lattice, M∗ , can be made very large, and the discussed solid phase should appear as a superlattice. For K-Li mixture this requires an increase in the mass of K by a factor of 20. For small fillings there is no interplay between the superlattice order and the shape of the underlying optical lattice, in contrast to the recently studied solid and supersolid phases in a triangular lattice with the filling factor of order one.17 The crystalline phase that we are describing is qualitatively different. In particular, it supports two branches of phonons with significantly different sound velocities, which can be used for the detection of the gas-crystal phase transition. The gaseous and solid phases of weakly bound molecules are actually metastable. The main decay channels are the relaxation of molecules into deep bound states and the formation of trimer states by one light and two heavy atoms. A detailed analysis

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of scattering properties of these molecules in free space has been made by Marcelis et al.,14 and here we focus on their stability in an optical lattice. For a large effective mass ratio M∗ /m, the relaxation into deep states occurs when a molecule is approached by another light atom and both light-heavy separa18 tions are of the order of the size of a deep state, Re κ−1 The released binding 0 . energy is taken by outgoing particles which escape from the sample. The rate of this process is not influenced by the optical lattice. We estimate the relaxation rate in the solid phase and near the gas-solid tran¯ −1 . At light-heavy separations r1,2 κ−1 the sition to the leading order in (κ0 R) 0 ¯ ˜ = B(κ−1 , R)ψ(r initial-state wavefunction reads: Ψ 1 , r2 ). Writing it as an antisym0 metrized product of wavefunctions (1), for the 2 × 3 case (κ−1 = a) we arrive at 0 ¯ 2 ) exp(−R/a). ¯ B ≈ (1/Ra The quantity W = B 2 Re6 is the probability of having both light atoms at distances ∼ Re from a heavy atom, and the relaxation rate is ν3D ∝ W . As the short-range physics is characterized by the energy scale ~2 /mRe2 , we restore the dimensions and write ¯ 2 ) exp(−2R/a), ¯ ν3D = C(~/m)(Re /a)4 (1/R

(18)

¯ −2 ≈ n. The coefficient C depends on a particular system and is ∼ 1 within where R an order of magnitude. The relaxation rate ν3D is generally rather low. For K-Li mixture where Re ≈ 50˚ A, even at na2 = 0.24 (see Fig. 1) the relaxation time exceeds 10 s for n = 109 cm−2 and a = 1600˚ A. In the 2 × 2 case, for the same n −1 and κ0 the probability W is smaller and the relaxation is slower. The formation of trimer bound states by one light and two heavy atoms occurs when two molecules approach each other at distances R . κ−1 0 . It is accompanied by a release of the second light atom. The existence of the trimer states is seen considering a light atom interacting with two heavy ones. The lowest energy solution of Eq. (3) for N = 2 is the gerade state (C1 = C2 ). Its energy + (R) introduces an effective attractive potential acting on the heavy atoms, and the trimer states are bound states of two heavy atoms in this potential. In an optical lattice the trimers are eigenstates of the Hamiltonian H0 = P −(~2 /2M∗ ) i=1,2 ∆Ri ++ (R12 ). In a deep lattice one can neglect all higher bands and regard Ri as discrete lattice coordinates and ∆ as the lattice Laplacian. Then, the fermionic nature of the heavy atoms prohibits them to be in the same lattice site. For a very large mass ratio M∗ /m the kinetic energy term in H0 can be neglected, and the lowest trimer state has energy tr ≈ + (L), where L is the lattice period. It consists of a pair of heavy atoms localized at neighboring sites and a light atom in the gerade state. Higher trimer states are formed by heavy atoms localized in sites separated by distances R > L. This picture breaks down at large R, where the spacing between trimer levels is comparable with the tunneling energy ~2 /M∗ L2 and the heavy atoms are delocalized. In the many-body molecular system the scale of energies in Eq. (10) is much smaller than |0 |. Thus, the formation of trimers in molecule-molecule “collisions” is energetically allowed only if the trimer binding energy is tr < 20 . Since the

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lowest trimer energy in the optical lattice is + (L), the trimer formation requires the condition + (L) . 20 , which is equivalent to κ−1 0 & 1.6L in the 2 × 3 case and κ−1 0 & 1.25L in the 2×2 case. This means that for a sufficiently small molecular size or large lattice period L the formation of trimers is forbidden. This allows one to study the low density – high mass ratio part of the phase diagram in Fig. 1 without worrying about the trimer formation and relaxation to deeply bound molecular states. At a larger molecular size or smaller L the trimer formation is possible. For finding the rate we consider the interaction between two molecules as a reduced 3-body problem, accounting for the fact that one of the light atoms is in the gerade and the other one in the ungerade state (C1 = −C2 ). The gerade light atom is integrated out and is substituted by the effective potential + (R). For the ungerade state the adiabaticity breaks down at inter-heavy separations R . κ−1 0 , and the ungerade light atom is treated explicitly. The wavefunction of the reduced 3-body problem satisfies the Schr¨ odinger equation [H0 − ~2 ∇2r /2m − E]ψ({R}, r) = 0,

(19)

where the energy E is close to 20 , {R} denotes the set {R1 , R2 }, and r is the coordinate of the ungerade atom. The interaction between this atom and the heavy ones is replaced by the boundary condition (2) on ψ. The 3-body problem can then be solved by encoding the information on the wavefunction ψ in an auxiliary ˜ 19 and representing the solution of Eq. (19) in the form: function f ({R}) X ˜ ({R})(G ˜ ˜ ˜ (20) ψ= χν ({R})χ∗ν ({R})f κν (r− R1)−Gκν (r− R2)), ˜ ν {R},

p where χν ({R}) is an eigenfunction of H0 with energy ν , and κν = 2m(ν −E)/~2 . For ν < E the trimer formation in the state ν is possible. This is consistent with imaginary κν and the Green function Gκν describing an outgoing wave of the light atom and trimer. We derive an equation for the function f in a deep lattice, where the tunneling energy ~2 /M∗ L2 |0 |. Then the main contribution to the sum in Eq. (20) comes 2 2 from the states ν for which |ν − p+ (R12 )| . ~ /M∗ L . The sum is calculated by expanding κν around κ(R12 ) = 2m(+ (R12 ) − E)/~2 up to first order in (ν − + (R12 ))/0 and using the equation (H0 − ν )χν = 0. The equation for f is then obtained by taking the limit r → R1 in the resulting expression for ψ and comparing it with the boundary condition (2). This yields X (21) [−(~2 /2M∗ ) ∆Ri + Ueff (R12 )]f (R1 , R2 ) = 0, i=1,2

where the effective potential Ueff (R12 ) is given by Ueff (R) =

λ(κ(R)) − Gκ(R) (R) ~2 κ(R) . m (∂/∂κ)[λ(κ(R)) − Gκ(R) (R)]

(22)

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At large distances Ueff ≈ U (R) + 20 − E, and for smaller R where + (R) < E, the potential Ueff acquires an imaginary part accounting for the decay of molecules into trimers. The number of trimer states that can be formed rapidly grows with the molecular size. Eventually it becomes independent of L and so does the loss rate. In this limit, we solve Eq. (21) for two molecules with zero total momentum under ¯ ≈ n−1/2 . We thus obtain the condition that f (R1 , R2 ) is maximal for |R1 −R2 | = R E as a function of the density and mass ratio, and its imaginary part gives the loss −2 rate ν for the many body system. Numerical analysis for 0.06 < nκ p0 < 0.4 and −2 50 < M∗ /m < 2000 is well fitted by ν ≈ (D~n/M∗ )(nκ0 ) exp(−J M∗ /m), with −2 2 D = 7 and J = 0.9 − 1.4(nκ−2 0 ) for the 2 × 3 case, and D = 10 , J = 1.4 − 2.8(nκ0 ) in the 2×2 case. One can suppress ν by increasing M∗ /m, whereas for M∗ /m < 100 and nκ−2 0 > 0.2 the trimers involving (light) Li atoms can be formed on a time scale τ . 1 s at densities n & 109 cm−2 . We explicitly check that the above predictions for the loss rate obtained from the solution of the two-molecule problem are not much altered in the many-body case. This is done in a perturbative way, averaging Im{Ueff (R)} over the DMC many-body ground state obtained with the potential Re{Ueff (R)}. 6. Concluding Remarks We have shown that the system of weakly bound molecules of heavy and light fermionic atoms can undergo a gas-crystal quantum transition. The necessary mass ratio is above 100 and the observation of such crystalline order requires an optical lattice for heavy atoms, where it should appear as a superlattice. A promising candidate is the 6 Li-40 K mixture as the light (Li) atom may tunnel freely in a lattice while localizing the heavy K particles to reach high mass ratios. A lattice with period 250 nm and K effective mass M∗ = 20M provide a tunneling rate ∼ 103 s−1 sufficiently fast to let the crystal form. Near a Feshbach resonance, a value a = 500 nm gives a binding energy 300 nK, and significantly lower temperatures should be reached in the gas. The parameters nκ−2 0 of Fig. 1 are then obtained at 2D densities in the range 107 − 108 cm−2 easily reachable in current experiments. In the vicinity of the transition such molecules are sufficiently stable. For densities n ∼ 108 cm−2 their lifetime is of the order of seconds. It is easy to see that fermionic dimers consisting of heavy bosonic and light fermionic atoms should undergo a similar gas-crystal transition. Indeed, the exchange repulsion mechanism is the same as long as the light fermions are identical. Moreover, the properties of the crystal phase should not be sensitive to the statistics of molecules as the crystal is characterized by their strong localization near the classical equilibrium positions. In particular, the harmonic approximation described in Sec. 4 should work for fermionic molecules as well as for bosonic. On the other hand, the exact quantitative description of fermions close to the transition by using the MC methods is rather challenging because of the sign problem.

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Returning to the bosonic 6 Li-40 K dimers in an optical lattice we would like to point to another interesting aspect of this system. For not very large effective mass ratios the dimers rapidly decay due to the formation of the bound states consisting of two heavy atoms localized at different lattice sites and a light atom delocalized between them. These peculiar trimers do not exist in free space and in the lattice they are long-lived as their decay into a 6 Li-40 K (or 40 K-40 K) deeply bound molecule and a free atom requires two 40 K atoms to be on the same lattice site, which is strongly suppressed by the Pauli principle. This indicates that optical lattices can become a powerful tool to study the few-body problem and quantum chemistry with ultracold gases. Acknowledgments This work was supported by the IFRAF Institute, by ANR (grants 05-BLAN-0205, 05-NANO-008-02, and 06-NANO-014-01), by the Dutch Foundation FOM, by the Russian Foundation for Fundamental Research, and by the National Science Foundation (Grant No. PHY05-51164). LKB is a research unit no.8552 of CNRS, ENS, and University of Pierre et Marie Curie. LPTMS is a research unit no.8626 of CNRS and University Paris-Sud. References 1. M.W. Zwierlein et al., Nature 435, 1047 (2005). 2. M. Greiner et al., Nature 426, 537 (2003); S. Jochim et al., Science 302; M. W. Zwierlein et al., Phys. Rev. Lett. 91, 250401 (2003); T. Bourdel et al., ibid. 93, 050401 (2004); G.B. Partridge, et al., ibid. 95, 020404 (2005). 3. D. S. Petrov, C. Salomon, and G. V. Shlyapnikov, Phys. Rev. Lett. 93, 090404 (2004); Phys. Rev. A 71, 012708 (2005). 4. D.S. Petrov, C. Salomon, and G.V. Shlyapnikov, J. Phys. B 38, S645 (2005). 5. C. Ospelkaus et al., Phys. Rev. Lett. 97, 120402 (2006). 6. In the 2D regime achieved by confining the light-atom motion to zero point oscillations with amplitude l0 , the weakly bound molecular states exist at a negative a satisfying the condition |a| l0 . See D.S. Petrov and G.V. Shlyapnikov, Phys. Rev. A 64, 012706 (2001). 7. D.R. Nelson and S. Seung, Phys. Rev. B 39, 9153 (1989). 8. W.R. Magro and D.M. Ceperley, Phys. Rev. B 48, 411 (1993). 9. H. Nordborg and G. Blatter, Phys. Rev. Lett. 79, 1925 (1997). 10. H. Bethe and R. Peierls, Proc. R. Soc. London, Ser. A 148, 146 (1935). 11. For a general reference on the DMC method see, e.g., J. Boronat and J. Casulleras, Phys. Rev. B 49, 8920 (1994). 12. G.E. Astrakharchik, J. Boronat, I.L. Kurbakov, Yu.E. Lozovik, Phys. Rev. Lett. 98, 060405 (2007). 13. L. Reatto and G.V. Chester, Phys. Rev. 155, 88 (1967). 14. B. Marcelis, S. J. J. M. F. Kokkelmans, G. V. Shlyapnikov, and D. S. Petrov, arXiv:0711.4632. 15. L.D. Landau and E.M. Lifschitz, Statistical Physics, Pergamon Press, 1980. 16. L. Xing, Phys. Rev. B 42, 8426 (1990).

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17. S. Wessel and M. Troyer, Phys. Rev. Lett. 95, 127205 (2005); D. Heidarian and K. Damle, ibid. 95, 127206 (2005); R.G. Melko et al., ibid. 95, 127207 (2005). 18. The relaxation involving one light and two heavy atoms is strongly suppressed as it requires the heavy atoms to approach each other and get to the same lattice site. 19. D.S. Petrov, Phys. Rev. A 67, 010703(R) (2003).

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LOCALIZATION AND GLASSINESS OF BOSONIC MIXTURES IN OPTICAL LATTICES T. ROSCILDE∗ , B. HORSTMANN, AND J. I. CIRAC Max-Planck Institute for Quantum Optics, Hans-Kopfermann-strasse 1, 85748 Garching, Germany ∗ E-mail: [email protected] In this paper we deal with the general subject of realizing disordered states in optical lattices by using an unequal mixture of fast and slow (or frozen) particles. We discuss the onset of Anderson localization of fast hardcore bosons when brought into interaction with the random potential created by secondary hardcore bosons frozen in a superfluid state. In the case of softcore bosons we discuss how localization phenomena, in the form of fragmentation of the mixture into many metastable droplets, intervene when trying to reach the equilibrium ground state of the system. Keywords: Ultra-cold atoms; optical lattices; Anderson localization; glassy behavior.

Ultracold bosons in optical lattices have the potential to realize fundamental and novel phases of correlated quantum matter;1 a unique advantage of optical lattices is that not only equilibrium states can be engineered by adiabatic preparation, but also out-of-equilibrium ones, which do not need to be eigenstates of the system Hamiltonian. If photon absorption/emission is low, the energy of the atoms is approximately conserved over significant time scales (comparable with those of the whole experiment), so that out-of-equilibrium states have a very slow decay; this is in contrast with the rapid relaxation of solid state systems due to the coupling between the various degrees of freedom and to the coupling with a thermal bath. In this paper we shortly review our recent activity on the out-of-equilibrium realization of quantum phases dominated by disorder in a mixture of bosons with unequal effective masses. The disorder is either explicitly introduced in the system by quenching the dynamics of one of the two species into a superposition of various random potential realizations,2,3 or spontaneously generated in the mixture via the trapping of the dynamics in metastable glassy states.4 Hence the use of unequal mixtures emerge as a viable route to create disorder in optical lattices, alternative to the optically generated randomness in laser-speckle potentials 5 and in incommensurate superlattices.6

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We investigate the two-boson Bose–Hubbard model in a one-dimensional optical lattice, with Hamiltonian Xh − Ja ai a†i+1 + h.c. − Jb bi b†i+1 + h.c. H= i

+

i Ua Ub ni,a (ni,a − 1) + ni,b (ni,b − 1) + Uab ni,a ni,b 2 2

(1)

where a, a† and b, b† are bosonic operators. Experimental realizations of such a system are currently offered by, e.g., mixtures of 87 Rb atoms in two different hyperfine states,7 or by heteronuclear mixtures of 87 Rb and 39 K.8 The above Hamiltonian has been proposed in Ref. 2,3 as a way to realize localized states with strong correlations. The fundamental idea is that of freezing the dynamics of the b bosons (i.e. of setting Jb = 0) after having prepared them in P a given state |Ψb i = {ni,b } c({ni,b })|{ni,b }i written as a superposition of Fock states |{ni,b }i, and to subsequently couple the two species of bosons, turning on the Uab interaction at time t = 0. For t > 0 the a bosons are then going to evolve in a quantum superposition of various random potentials, each corresponding to a different Fock state component of |Ψb i; instantaneous expectation values related to the a bosons are given by X |c({ni,b })|2 hΨa (t; {ni,b })|h{ni,b }| A |Ψa (t; {ni,b })i|{ni,b }i (2) hAi(t) = {ni,b }

where |Ψa (t; {ni,b })i is the wavefunction for the a bosons evolved in the random potential Vi = Uab ni,b . This preparation protocol can be experimentally realized with the use of spin-dependent optical lattices.

200 0.8 100 0.6

i−i0

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0 0.4 −100 0.2 −200 0

20

40 tJ 60

80

100

120

0

a

Fig. 1. Left: time-dependent density profile of Na = 35 hardcore bosons expanding from a perfect Mott-insulating state in the the random potential created by the b frozen bosons (U a b = 0.5Ja ). Right: time evolution of the correlator ρi0 ,i0 +r = ha†i0 ai0 +r i calculated with respect to the center of the system i0 , for a system of Na = 100 bosons starting from a superfluid state in the parabolic trap (Vt = 2.5 × 10−5 Ja ).

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We have studied the dynamical onset of localization of a bosons initially prepared in a parabolic trap, and then set free to propagate in the random potential created by the frozen b bosons. Taking the hardcore-repulsion limit for both species, Ua , Ub → ∞, in D = 1 we can use Jordan–Wigner diagonalization9,10 to exactly build the state |Ψb i in which we freeze the b particles, and the initial state |Ψa (t = 0)i for the a particles, via the factorized ground state for the Hamiltonian Eq.(1) with Uab = 0. Moreover the same exact diagonalization technique can be used to calculate the dynamics of the system as in Eq.(2), where the averaging over the disorder distribution |c({ni,b })|2 is done by Monte-Carlo importance sampling. The correlated disorder potential associated with the Fock-state components of the superfluid state |Ψb i is seen to lead to clear Anderson localization effects. The exP pansion of the a bosons, initially confined in a parabolic trap Vt i i2 ni,a , is seen to stop for any strength of the coupling Uab which we could numerically investigate, and the steady-state density profile features exponentially decaying tails typical of Anderson localization.3 When starting from a superfluid state in the trap, the a bosons evolve from a state with power-law decaying correlations ha 0 ar i ∼ r−1/2 at t = 0 into a state with exponentially decaying correlations, characteristic of an Anderson insulator (see Fig. 1, right panel). If the initial state is a perfect Mott state, the phenomenon of quasi-condensation at finite momenta k = ±π/2, reported in absence of disorder,10 is completely suppressed by the random potential (see Fig. 1, left panel). Hence the realization of a genuine Anderson-localized state as the steady state of the evolution emerges as a very robust property of this protocol. ni,a after 500 MCS ni,b after 500 MCS

3.5

5

-0.8

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phase -0.9 separation -1

L=48 L=72 L=96

-1.1 0

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ni,a after 3*10 MCS

2.5 2 1.5 1 0.5

0.4

0

10

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i

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Fig. 2. Left: energy per site of the quantum emulsion states (obtained after thermal annealing) as a function of the phase interface A; here Ua = Ja , other parameters are as in the text. Right: instantaneous density snapshots during a QMC simulation on a L = 72 system (U a = 2Ja , other parameters as in the text).

Localization phenomena in unequal mixtures can also be obtained in strongly metastable states of the Hamiltonian Eq.(1). Relaxing the hard-core constraint introduced in the previous section, we consider the case of strong a-b and b-b interaction, and of tunable a-a interaction. When Ua Ua b, Ub the ground state of the system is phase-separated, but all states in the subspace with no sites simultane-

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ously occupied by an a and a b boson (later referred to as Ha.xor.b ) are strongly metastable, given that escaping from those states requires e.g. an a particle to overlap with a b particle, namely to overcome a barrier ∼ Uab . Restricting the dynamics of the system to the subspace Ha.xor.b implies to consider only multiple hopping events: assuming e.g. the fillings na = 1 and nb = 1/2, a triple hopping event with amplitude Jeff ∼ Ja2 Jb /(Uab − Ua )2 is required to swap the occupation of two neighboring sites, which leads to a considerably slow dynamics in the system. We consider the b bosons to have a higher effective mass than the a ones, Jb Ja ; in addition, if Ja < Uab − Ua , the strong interactions among the two species further slow down the dynamics, and Jeff < Jb . Hence metastable localization effects can take place even if the b bosons are not fully frozen. In particular a quantum Monte Carlo (QMC) study4,11 for the parameter set Jb = 0.2Ja , Uab = Ub = 5Ja and Ua . 4Ja , shows that the system has the marked tendency to be trapped in metastable states characterized by the presence of several non-overlapping droplets of the two species, similar to an emulsion of two immiscible classical fluids, and hence dubbed quantum emulsion states. Thermal annealing of the system to low temperatures systematically fails to reach the true ground state and gets trapped in metastable quantum emulsion states which persist over hundreds of thousands of composite QMC steps (see Fig. 2, right panel). Recording the energy of the metastable states reveals its roughly linear dependence on the number of droplets (or phase interface, see Fig. 2, left panel): hence the system displays quasi-degeneracy upon permutation of the droplets, showing an exponential proliferation of the quantum emulsion states when the system size is increased. An analogous picture emerges when using QMC quantum annealing,12 namely changing the Hamiltonian parameters at low temperature to schematically mimic an experimental sequence of loading of the two species in an optical lattice and subsequently lowering the interaction Ua through a Feshbach resonance. The quantum emulsion phase is hence a glassy low-temperature phase characterized by exponentially many metastable states masking a phase-separated ground state: coming from high temperatures (or from a weakly interacting limit at low temperature), the system can be “supercooled” (or “quantum-quenched”) in an amorphous glassy structure, with exceedingly long relaxation times. A characterization of this behavior in terms of a quantum glass transition is currently in progress. Acknowledgment This work is supported by the E.U. through the SCALA integrated project. References 1. 2. 3. 4.

I. Bloch, J. Dalibard, and W. Zwerger, arXiv:0704.3011 (2007). B. Paredes et al., Phys. Rev. Lett. 95, 140501 (2005). B. Horstmann, J. I. Cirac, and T. Roscilde, arXiv:0706.0823 (2007). T. Roscilde and J. I. Cirac, Phys. Rev. Lett. 98, 190402 (2007).

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5. J. Lye et al., Phys. Rev. Lett. 95, 070401 (2005); D. Clément et al., Phys. Rev. Lett. 95, 170409 (2005); C. Fort et al., Phys. Rev. Lett. 95, 170410 (2005); T. Schulte et al., Phys. Rev. Lett. 95, 170411 (2005). 6. L. Fallani et al., Phys. Rev. Lett. 98, 130404 (2007). 7. O. Mandel et al., Phys. Rev. Lett. 91, 010407 (2003). 8. J. Catani et al., arXiv:0706.2781 (2007). 9. E. Lieb, T. Schultz, and D. Mattis, Ann. Phys (NY) 16, 406 (1961). 10. M. Rigol and A. Muramatsu, Phys. Rev. A 70, 031603 (2004); Phys. Rev. Lett. 93, 230404 (2004). 11. O. F. Sylju˚ asen, Phys. Rev. E 67, 046701 (2003). 12. G. E. Santoro et al., Science 295, 2427 (2002).

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SCATTERING OF A SOUND WAVE ON A VORTEX IN BOSE-EINSTEIN CONDENSATES P. CAPUZZI Departamento de F´ısica, Universidad de Buenos Aires, Buenos Aires, RA-1428, and Consejo Nacional de Investigaciones Cient´ıficas y T´ ecnicas, Argentina E-mail: [email protected] F. FEDERICI∗ and M. P. TOSI† NEST-CNR-INFM and Classe di Scienze, Scuola Normale Superiore, Pisa, I-56126, Italy ∗ E-mail: [email protected] † E-mail: [email protected] We investigate the scattering of sound wave perturbations by vortices in trapped Bose– Einstein condensates. Using a variational approach, we show that the energy barrier between the ground state and a state containing an axis-symmetric vortex can be surpassed by scattering with density perturbations. The transfer of angular momentum from the condensate to the perturbation can be made total for suitably chosen density perturbations carrying an energy of about 10% of the ground-state energy. Keywords: Bose–Einstein condensation; Sound wave propagation; Superradiance.

1. Introduction The scattering of sound wave perturbations from vortex excitations in Bose–Einstein condensates (BEC) is investigated following up recent studies on scattering from hydrodynamic vortices in the classical regime.1,2 Such studies indicate that in the perturbative limit sound wavepackets can extract a sizeable fraction of the vortex energy through a mechanism of superradiant scattering.3–5 The possibility that substantial angular momentum transfer may persist even in the non-perturbative quantum regime described by the Gross–Pitaevskii functional is what this work focuses on. For this purpose, we resort to a simple variational wavefunction to describe the steady states of a BEC in a spherical trap as a function of the number of particles. The variational approach allows us to analyze the energy barrier between the ground state and an axis-symmetric quantized vortex giving an energy difference between these states of less than 10%. Extending the variational wavefunction to time-dependent scenarios we found that a sizable fraction of the angular momentum can be extracted by an impinging wavepacket carrying enough energy. The work is organized as follows. In Sec. 2 we perform the variational calculation

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for the steady states of the BEC and study its energy landscape as a function of the variational parameters. In Sec. 3 we solve the dynamics of the system and analyze the extraction of angular momentum by the impinging wavepacket. Finally, Sec. 4 offers a brief summary and outlook. 2. Steady States and Energy Landscape At zero temperature, a dilute BEC of atoms is well described by the Gross–Pitaevskii (GP) energy functional (see, e.g, Ref. 6 for a review) 2 Z ~ g E[ψ] = d3 r (1) |∇ψ|2 + Vext (r)|ψ|2 + |ψ|4 2m 2

where ψ(r) is the condensate wavefunction, Vext = mω 2 r2 /2 is the harmonic confining potential with angular frequency ω, and g = 4π~2 a/m is the interaction strength with a the s-wave scattering length and m the atom mass. Qualitative results for the energy eigenstates can be obtained by parametrizing ψ in the following Gaussian form mω 2 r2 x + iy − N 1/2 mω 1/2 cos τ + sin τ e 2~ b2 (2) ψ(r) = 1/2 π~ aho b b where aho = (~/(mω))1/2 is the oscillator length. This yields the energy E=

~ω 1 mω 3/2 1 1 √ 3 2N02 + 4N0 N1 + N12 (3N0 + 5N1 )(b2 + 2 ) + g 4 b ~π 8 2b

(3)

where N0 = N sin2 τ and N1 = N cos2 τ are the number of particles in the ground state and axis-symmetric vortex, respectively. The energy functional possesses three local extrema: the ground state ψ0 at τ = 0, the vortex ψ1 at τ = π/2, and a maximum around τ ' 0.8 representing the energy barrier between the ground state and the vortex. This barrier is better observed in Fig. 1 by evaluating Eq. (3) for the superposition ψ(r) = N 1/2 [sin τ ψ0 (r) + cos τ ψ1 (r)]. Furthermore, expression (3) at high coupling yields a value of the barrier of about 10% E0 , with E0 the groundstate energy. This suggests that a transition from ψ1 to ψ0 could be possible with a moderate energetic cost. 3. Scattering of a Wavepacket The study of the dynamics of the condensate within a variational approach starts with the construction of the Lagrangian8 Z ~2 ∂ψ ∂ψ ∗ g i~ ψ∗ − −ψ |∇ψ|2 − Vext (r)|ψ|2 − |ψ|4 d3 r (4) L[ψ, t] = 2 ∂t ∂t 2m 2

and a model for the time-dependent wavefunction. In order to include a mechanism that transfers atoms from the vortex to the ground state we study the impact of a localized wavepacket onto the condensate as expressed by the following wavefunction ψ(r, t) = A(t)ψ0 (r) + B(t)ψ1 (r) + C(t) eik·r φ(r − r0 ).

(5)

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N = 105 N = 104

E[b0 , d1 , τ ]/E0

1.2

1.1 W

E1 1

E0

−π

0 τ

−π/2

π/2

π

Fig. 1. Condensate energy E (in units of E0 ) as function of τ for two values of N . In the calculation we took ω/(2π) = 11.7 Hz and the value of a and m of 87 Rb atoms.7

Here A(t), B(t), and C(t) are complex amplitudes and r0 (t) and k(t) real functions of time. For the last term in Eq. (5), representing particles around r0 moving with average momentum ~k, we take a narrow Gaussian packet of fixed width b p , φ(r) = exp(−r 2 /b2p ). The time-dependent parameters entering Eq. (5) then satisfy the coupled Euler–Lagrange equations d ∂L ∂L = , for α = A, B, C, r0 and k. (6) dt ∂ α˙ ∂α The trajectory of a wavepacket carrying Np particles and impinging on a vortex with 105 atoms is displayed in Fig. 2 for Np /N ' 8 × 10−2 . Due to the confinement, the vortex scatters periodically with the wavepacket and exchanges angular momentum with the same periodicity. The change in the area of the orbit reflects the change in the angular momentum carried by the wavepacket, the smaller the area the smaller the angular momentum. A direct calculation of Lz from the variational parameters shows that the angular momentum of the wavepacket can be fully transferred to the condensate during the bp /aho = 1, k0 aho = 5.5

bp /aho = 1, k0 aho = 10

bp /aho = 3, k0 aho = 5.5

bp /aho = 3, k0 aho = 10

10

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−10 −10

0

x/aho

10

−10

0

x/aho

10

−10

0

x/aho

10

−10

0

10

x/aho

Fig. 2. Trajectories of the center of the incident wavepacket r0 (in units of aho ) for the period ωt = 0 − 50. Each panel corresponds to different initial velocities ~k0 /m and wavepacket widths bp as indicated in the plot. The dashed circles mark the radius of the BEC.

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ep = 10% ep = 3%

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Lz /¯h

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0 0

5

10

15

20

25

ωt (a)

30

35

40

45

50

ψ0 π/4 τ

0

π/2

(b)

Fig. 3. (a) Angular momentum Lz (in units of ~) of the BEC (solid lines) and wavepacket (dashed lines) per total number of particles as function of time t (in units of 1/ω) for two values of e p (in units of E0 ). (b) Energy landscape Eq. (3) together with the values of the energy of the BEC during the time evolution (see text).

scattering event. We found that for wavepacket energies, ep , bigger than about 10% of E0 , the angular momentum is fully transferred for long periods of time. For lower energies, instead, the wavepacket scatters from the condensate without any substantial exchange of angular momentum. This behavior is illustrated in Fig. 3 for ep = 10% and 3%. In the right panel of the figure we show the energy landscape Eq. (3) together with the values of energy of the BEC when the wavepacket is at distances larger than the width of the condensate, that is, when the distinction between the BEC and wavepacket is clear. The analysis of the evolution of L z as a function of ep for several wavepacket widths and number of particles indicates that as ep increases from zero the angular momentum decreases, vanishing for ep around 8% irrespectively of the number of particles.9 Therefore, it seems possible to choose k0 and thereby tune ep so that all the angular momentum is extracted from the vortex.

4. Summary and Concluding Remarks Using a variational approach we have shown that the energy barrier associated with the crossover from the metastable state of the BEC with an axis-symmetric vortex to its ground state is about 10% of the ground-state energy, for system parameters taken from recent experiments. The dynamical study of the scattering between the vortex line and a wavepacket with several initial velocities and widths has been carried out by means of the Euler–Lagrange equations for the variational parameters describing the system. From the trajectories of the wavepacket and the computation of the angular momentum of the BEC it is clear that, after every scattering against the condensate, there is a change in the angular momentum of the wavepacket that lasts for an appreciable time during the evolution, if the wavepacket carries an energy bigger than the energy barrier. By removing the wavepacket after the relevant scattering event, to prevent it from scattering back and forth, it could be possible to stabilize an energetic transition of the system from the vortex state to

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the ground state, opening up the possibility of a transition between two states of different angular momentum. From the outcome of our investigation our interest is immediately drawn to the study of the exact dynamics of the system, as described by the time-dependent Gross–Pitaevskii equation, in order to assess to what extent the predictions of the variational model may persist. We can confidently anticipate that the main differences are due to the diffusion of the wavepacket and the condensate, as well as the excitation of other surface collective modes. These effects will influence the dynamics of the system, in particular by subtracting some of the energy supplied by the wavepacket. Acknowledgments P. C wishes to acknowledge support from CONICET, Argentina through grant PIP 5138/05 and to SNS, Pisa where the initial part of this work was performed. References 1. 2. 3. 4. 5. 6.

F. Federici, C. Cherubini, S. Succi, and M. P. Tosi, Phys. Rev. A 73, 033604 (2006). C. Cherubini, F. Federici, S. Succi, and M. P. Tosi, Phys. Rev. D 72, 084016 (2005). Ya. B. Zel’dovich, JETP Lett. 14, 180 (1971); ibib, Sov. Phys. JETP 35, 1085 (1972). S. Basak and P. Majumdar, Class. Quant. Grav. 20, 2929 (2003); ibid., 3907 (2003). E. Berti, V. Cardoso, and J. P. S. Lemos, Phys. Rev. D 70, 124006 (2004). F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71, 463 (1999). 7. K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, Phys. Rev. Lett. 84, 806 (2000). 8. A. A. Svidzinski and A. L. Fetter, Phys. Rev. Lett. 84, 5919 (2000). 9. P. Capuzzi, F. Federici, and M. P. Tosi, in preparation.

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STATIC PROPERTIES OF A SYSTEM OF BOSE HARD RODS IN ONE DIMENSION F. MAZZANTI Departament de F´ısica i Enginyeria Nuclear, Universitat Polit` ecnica de Catalunya, Comte Borrell 187, E-08036 Barcelona, Spain E-mail: [email protected] G. E. ASTRAKHARCHIK, J. BORONAT and J. CASULLERAS Departament de F´ısica i Enginyeria Nuclear, Universitat Polit` ecnica de Catalunya, Campus Nord B4-B5, E-08034 Barcelona, Spain We study the groundstate properties of a system of Bose Hard Rods of length a in one dimension by means of a Monte Carlo simulation. Since the analytical expression of the ground state wavefunction is known, the result of the calculation is exact. We discuss the behaviour of the static structure factor S(k), the pair distribution function g(z) and the momentum distribution n(k), and compare them to existing asymptotic expansions valid for Luttinger liquids at large distances.6 The analysis of S(k) reveals that two completely different regimes exist, characteristic of a solid-like (high density) and a gas-like (low density) phases. Furthermore, exact analytical values for S(k) at the momenta kj = 2πnj/a with n the density and j an integer, are derived. The one-body density matrix presents a power-law decay at large distances that turns into a divergent √ behaviour in n(k → 0) for densities lower than a critical value nc a = 1 − 1/ 2, thus stressing the presence of a Bose quasi-condensate. Keywords: Hard Rods; Monte Carlo; static properties; quasicondensate.

1. Introduction Correlated (quasi)-one dimensional (1D) systems of bosons and fermions have received great attention in the last years due to recent and important experimental progress.1–4 1D system can be realized confining the radial motion of a 3D trapped cloud of atoms to zero-point oscillations. In 1998, Olshanii5 showed that the scattering lengths of the 3D interatomic potential and the confining potential determine the scattering length of the equivalent 1D system. In this context a system of Hard Rods, characterized by the rod size a, becomes a universal model that is able to describe 1D interacting systems at low densities where only the scattering length of the potential is relevant, and a model of strongly repulsive interaction at high densities. Hard rods interact through the pairwise potential V (zij ) = +∞ for |zij |< a and 0 otherwise, thus being the equivalent of a hard spheres potential in 3D. For this

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system the exact bosonic ground-state wave function is known and reads7 1 1 Ψ0 (z1 , z2 , ..., zN ) = √ det √ exp(ip0k xk ) . 0 N! L

(1)

In this expression, L0 = L − aN is the unexcluded length, {p0k = 2πnk /L0 } are a set of quantum numbers with nk ∈ [−N, +N ], and xk = zk − (k − 1)a are the socalled rod coordinates corresponding to a given ordering of the true particle positions z1 < z2 − a < z3 − 2a < · · · < zN − a(N − 1). As in the 3D case of hard spheres, the scattering length of the hard rod potential equals the size of the rod, a1D = a. In this work we use the Monte Carlo method to analyze the most relevant groundstate properties of a system of N ≤ 300 hard rods in a box of size L with periodic boundary conditions, by sampling the exact wave function of Eq. (1). We first analyze the static structure factor S(k) = hΨ0 | ρ†k ρk | Ψ0 i/N , with P ikzj ρk = N the density fluctuation operator. There are three main conclussions j=1 e one can draw about this function. First of all, the systems is a realization of a Luttinger liquid and therefore the low k behavior of S(k) is linear with a coefficient that is inversely proportional to the sound velocity c of the medium. The analytical expression of c is known for this system and one finds S(k → 0) ≈ (1 − an)2 |k|/2πn. On the other hand and for a given particle ordering, the density fluctuation operator PN becomes ρk = j=1 eikxj when k is a multiple of 2π/a. In this case, the a factors in the change to rod coordinates {zk } → {xk } become irrelevant, and S(k) equals the corresponding value of the static structure factor of the 1D free Fermi gas (FFG) at the rod density n0 = n/(1 − an) 1−an 2πn 2πn |k| for |k| ≤ 1−an (2) SF F G (k) = 1 otherwise . Finally and with the wave function in Eq. (1) it is easy to see that the most probable configurations are those were all rods are equally spaced. Taking into account that for hard rods S(k) can be cast in the form S(k) = 1 + 2(N − 1)

N −1N −i Z X X i=1 j=1

ΩN

dxN cos [k(xi+j,i + ja)] Ψ20 ,

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one realizes that the contribution to S(k) becomes maximal at the discrete momenta kj = 2πnj with j an integer. Results for S(k) at different densities are shown in Fig. 1. Our simulations indicate that the height of the peaks increase with the number of particles in the simulation. This dependence can be understood by looking at the asymptotic expansion of the pair distribution function, the Fourier transform of S(k), which admits, for a Luttinger liquid and according to Haldane,6 the following asymptotic expansion valid when |z| n−1 ∞ X η cos(2πn m z) g(z) = 1 − + Am , (2πnz)2 m=1 (n|z|)m2 η

(4)

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Fig. 1. S(k) at particle densities na = 0.4, 0.6 and 0.8 (upper, middle and lower cures at low k). Notice the logarithmic scale up to k/2πn = 1. Solid line: phononic behavior. The crosses correspond to the exact values from Eq. (2), for na = 0.8 and ki = 2πi/a, i = 1, 2, 3, 4. Inset: height of the first peak as a function of the total number of particles (symbols) for densities na = 0.6, 0.7 and 0.8 (lower, middle and upper curves, respectively). The solid lines show the best fit of the form 2 2 Am N 1−2m (1−na) for k = 2πnm and m = 1 corresponding to the first peak in S(k).

where η = 2K and K = π~n/mc is the Luttinger parameter. In this expression, the coefficients Am depend both on the density and the system, while for hard rods η = 2(1 − an)2 . The FT of this expression reveals that the height of the m2 th peak follows a power-law of the form N 1−m η . The inset in Fig. 1 shows the dependence of the height of the first peak of S(k) as a function of the number of 2 particles for three different densities. The N 1−m η dependence indicates that only a finite number of peaks located at km = 2πnm exist. With η = 2(1 − an)2 this relation predicts a number of peaks m < 3.5 at na = 0.8, and we find only three peaks in our simulation whose height increases with the number of particles. The linear behavior S(k) ∝ N at the peak, characteristic of 3D crystals, is recovered asymptotically when na → 1. All these facts suggest that a packing order, resulting from the combined effect of particle correlations and the reduced dimensionality, shows up at high densities, thus manifesting the existence of a quasi-solid phase. At low density these effects, although present, are much less evident, as the peaks are washed out at na 1 and S(k) approaches the simple structure corresponding to a 1D free Fermi gas with the density n0 = n/(1 − an) as reported in Eq. (2). In this sense, the system of hard rods clearly presents different regimes, behaving as a quasi-solid and a quasi-gas at high and low densities, respectively. The next quantity analyzed is the momentum distribution n(k) which describes the occupation of each single-particle state of momentum k. This function is the Fourier transform of the one-body density matrix n1 (z), which for a Luttinger liquid admits also an asymptotic expansion of the form6 ∞ X 1 cos(2πn m z) n1 (z) = Bm 1/η n (n|z|) (n|z|)m2 η m=0

(5)

written in terms of the model-dependent factors Bm and the Luttinger parameter η = 2K. Figure 2 displays n(k) for three different densities. The large z power law decay of the one-body density matrix makes the low k behavior of n(k) depend

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Fig. 2. Momentum distribution n(k) of the hard rods system at the densities na = 0.2, 0.4 and 0.6 (upper, middle and lower curves). The inset shows the product k 1−1/η n(k) at na = 0.1 (squares) and na = 0.2 (stars). 2 on η. For √ hard rods η = 2(1 − na) and that defines a critical density nc a = 1 − 1/ 2 ≈ 0.29 that separats two different regimes. At densities n < nc the 2 momentum distribution presents an infrared divergence of the form | k |1/2(1−na) −1 , as can be checked by inspection of the Fourier transform of the m = 0 term of Eq. (5). At these densities n(k = 0) diverges while there is no long range order in the one-body density as stressed by the fact that n1 (|z| → ∞) does not saturate to a finite value in Eq. (5). This divergence can then be understood as the manifestation of a Bose quasi-condensate. A similar analysiss of the mq= 1 term indicates that n(k) √ presents a kink at k = 2πn for all densities na ≤ 1 − (3 − 5)/2 ≈ 0.56. Rusults for the momentum distribution at three different densities are shown in Fig. 2, where both the divergence at k = 0 and the kink at k = 2πn can be appreciated. In summary, we have carried out a complete study of the most relevant oneand two-body correlation functions of a system of Bose hard-rods at T = 0. We find two distinct regimes where the system behaves as a gas (low density) and as a solid (large density), without any trace of a phase transition in the energy. The quasi-solid regime is characterized by the presence of macroscopic peaks in the static structure factor. The one-body density matrix at large distances decays following a power law that leads to a divergence in the low density momentum distribution at k = 0. This divergence can be understood as the manifestation of a Bose–Einstein quasi-condensate.

References 1. 2. 3. 4. 5. 6. 7.

B. Paredes et al., Nature 429, 277 (2004). I. Bloch, Nature Phys. 1, 23 (2005). H. Moritz et al, Phys. Rev. Lett. 91, 250402 (2003). S. Richard et al., Phys. Rev. Lett. 91, 010405 (2003). M. Olshanii, Phys. Rev. Lett. 81, 938 (1998). F. D. M. Haldane, Phys. Rev. Lett. 47, 1840 (1981). T. Nagamiya, Proc. Phys. Math. Soc. Jpn. 22, 705 (1940).

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A SCENARIO FOR STUDYING OFF-AXIS VORTICES IN BOSE-EINSTEIN CONDENSATES Dora M. JEZEK, Horacio M. CATALDO, and Pablo CAPUZZI Departamento de F´ısica, Facultad de Cs. Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina and Consejo Nacional de Investigaciones Cient´ıficas y T´ ecnicas, Argentina We present a scenario for studying the dynamics of vortices in Bose–Einstein condensates. We construct a trapping potential that can sustain metastable off-axis vortices and propose a simple phase-imprinting method to numerically generate them. In contrast to previous works, we do not use a rotating frame or a centrifugal potential for their creation. Our method offers a good control in the number and location of the generated vortices. We include an analysis of the vortex energy as a function of its position. Keywords: Bose–Einstein condensates; vortices.

Vortices are central in understanding superfluids. In Bose–Einstein condensates (BECs) an enormous amount of work, both theoretical and experimental, has been devoted to studying their structure, stability and dynamics.1–3 Since quantized vortices in BECs were first experimentally produced, many techniques were developed for their experimental generation.4–6 In recent experiments, alternative trapping potentials, quadratic plus quartic polynomial traps, have been constructed in order to obtain vortex lattices in fast rotating condensates.7,8 From a theoretical point of view the numerical generation of vortices has not been an easy task. Vorticity has been obtained by using rotating frames or a centrifugal potential (CP) term in the energy.1–3 A condensate with the usual parabolic density profile, arising from an harmonic oscillator confinement, presents the drawback that off-axis vortices spiral away in presence of dissipation. This leads to the problem of finding a non-rotating trapping potential that can sustain a metastable off-axis vortex. In this work we present a polynomial trapping potential that can sustain such off-axis vortices, and also we describe a simple numerical phase imprinting (NPI) method for obtaining such states. With this trapping potential the system exhibits a vortex energy barrier in the border of the condensate, which prevents the vortex to spiral away. We have mathematically modeled a polynomial function for the potential, which

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written in cylindrical coordinates reads

Vtrap (r, z) =

1 r2 (r − r1 ) (r − r2 ) m ωr2 + ωz2 z 2 2 r1 r2

(1)

where ωr and ωz are radial and axial angular frequencies, respectively. As a function of r this trapping potential has two zeroes at r1 and r2 , a local maximum at R− < r1 , and two local minima at r = 0 and R+ with r1 < R+ < r2 . The minimum of the potential at R+ gives rise to a maximum in the ground state density that, as we shall see, generates a vortex-energy barrier, which prevents the vortex to spiral away from the condensate. On the other hand, the local potential maximum generates a local minimum in the density of particles, which in turn yields a local minimum of the vortex energy. p We have used a trapping potential with (r1 , r2 ) = (10, 18) (in units of aho =6 ~/(mωr )), which yields R− = 6 and R+ = 15. Our system is formed by N = 10 atoms of 87 Rb whose scattering length is a = 98.98a0, where a0 is the Bohr radius. We have chosen ωr /(2π) = 100 Hz and ωz /(2π) = 520 Hz, in order to obtain a pancake-shaped condensate. A ground-state density profile, obtained by solving the Gross–Pitaevskii (GP) equation, is shown in Fig. 1(a). The number of particles is large enough to consider that the condensate in the (x, y) plane is in the Thomas– Fermi (TF) regime. Depending on the values of r1 and r2 , one can model different shapes of densities, and thereby change the position of the critical points and height of the energy barrier. The numerical phase imprinting method minimizes the standard GP energy functional starting from a convenient guess. As a starting point for the minimization, we choose a smooth real function ψ0 (r), preferably the ground-state wavefunction,

600

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Fig. 1. (a) Density profiles of the ground state (solid line) and the stable vortex (dashed line) at y = 0 and z = 0. (b) Vortex energy Ev as a function of the position of the core r0 at the plane z = 0.

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and construct the state (x − xk ) + i(y − yk ) ψ0 (r), ψ 0 (r) = p (x − xk )2 + (y − yk )2

(2)

p with x2k + yk2 < R+ . This state can be written as ψ 0 = eiϕk ψ0 where ϕk is the azimuthal angle around the axis (xk , yk , z), and thus, the operation we have performed on ψ0 (r) represents a phase imprint to the original state. Therefore, while the density profiles of ψ0 and ψ 0 are the same, the state ψ 0 has an imprinted velocity field v = (~/m) ∇ϕk . We use ψ 0 as the initial guess of the energy minimization performed using a conjugate gradient technique.9 The program in a few iterations generates a density hole around the vorticity line, maintaining the original circulation. In Fig. 2 we show surfaces of constant density for two vortex states obtained using this method. In the calculations of this particular figure we have changed the ωz frequency to ωz = ωr in order to better illustrate the shape of the vortices. In Fig. 2(a) we show a vortex which was obtained after a few iterations from a vorticity line generated near (14, 0, z). It may be seen that the method allows the vortex to bend in order to minimize the energy, contrary to what would happen in a CP calculation. As the minimization proceeds the vortex moves towards an energy minimum and the state finally converges to the stable vortex, whose vorticity line is at (6, 0, z). This final configuration is displayed in Fig. 2(b). The density profile of the stable vortex compared to the ground-state one is shown in Fig. 1(a). It may be seen that the vortex is located around the minimum of the ground-state density. We have computed the energy of a state with a single vortex Evs from the GP energy functional by means of both the CP and the NPI methods. In Fig. 1(b) we show the vortex energy defined as Ev = Evs − E0 , where E0 is the groundstate energy. The points along the NPI curve were obtained from three different runs starting from states with imprinted vorticities next to r = 0, and on the left and right of R+ . Once the vortex core was generated we evaluated the energy as a function of its position, in the plane z = 0. The CP curve was obtained by minimizing the GP energy for a real wavefunction with the addition of a centrifugal potential around the axis (x0 , y0 , z). It is worthwhile noticing from Fig. 1(b) that the vortex energy Ev appreciably differs from the kinetic energy Ek = N mhv 2 i/2 arising from the vortex velocity field. Here it is important to remark that the energy

(a)

(b)

Fig. 2. (a) Surfaces of constant density for a vorticity line that crosses the z = 0 plane through the point (14, 0, 0). (b) Same as (a) for the stable vortex whose vorticity line is at (6, 0, z).

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Ek is commonly associated to the total energy of a vortex (see, for example Ref. 10, and references therein). We found that the vortex energy is well approximated by the following expression: Ev ∼ ρ0 (r0 , 0) F ,

(3)

where ρ0 (r0 , 0) denotes the density of the ground-state at r = r0 , z = 0, and the global factor F is utilized as an adjustable parameter. Recall that the ground-state density as a function of r exhibits a maximum near the border of the condensate. So, according to Eq. (3), such a maximum gives rise to a maximum of the energy, which corresponds to the barrier that forbids the vortex to spiral away from the condensate. Figure 1(b) shows the curve (solid line) obtained from Eq. (3) with the factor F = 30 fitted to the energy minimum at r0 = 6. Note that the numerical calculations are well reproduced by the estimate (3) using an appropriate value of F . We want to mention that the formulae (3) is similar to that given e.g. in Ref. 10 for the kinetic energy Ek . Finally, we may see that near the maximum R+ , the NPI method provides a lower energy than the other approaches due to the possibility this method offers in bending vortices. In conclusion, we have proposed a scenario that we hope will stimulate further theoretical and experimental investigations on the dynamics of vortices in nonrotating BECs. Acknowledgment This work has been performed under Grant PIP 5409 from CONICET, Argentina. References 1. A. L. Fetter and A. A. Svidzinsky, J. Phys.: Condens. Matter 13, R135 (2001). 2. C. J. Pethick and H. Smith, Bose–Einstein Condensation in Dilute Gases (Cambridge University Press, Cambridge, 2002). 3. L. Pitaevskii and S. Stringari, Bose–Einstein Condensation (Clarendon Press, Oxford, 2003). 4. M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C.E. Wieman, and E. A. Cornell, Phys. Rev. Lett. 83, 2498 (1999). 5. B. P. Anderson, C.A. Regal, D. L. Feder, L. A. Collins, C. W. Clark, and E. A. Cornell, Phys. Rev. Lett. 86, 2926 (2001). 6. A. E. Leanhardt, A. Gorlitz, A. P. Chikkatur, D. Kielpinski, Y. Shin, D. E. Pritchard, and W. Ketterle, Phys. Rev. Lett. 89, 190403 (2002). 7. V. Bretin, S. Stock, Y. Seurin, and J. Dalibard, Phys. Rev. Lett. 92, 050403 (2004). 8. S. Stock, B. Battelier, V. Bretin, Z. Hadzibabic, and J. Dalibard, cond-mat/0603792. 9. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 77: The Art of Scientific Computing (Cambridge University Press, Cambridge, 1992). 10. D. E. Sheehy and L. Radzihovsky, Phys. Rev. A 70, 063620 (2004).

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NUCLEAR AND SUBNUCLEAR PHYSICS

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STRANGENESS NUCLEAR PHYSICS ANGELS RAMOS Departament d’Estructura i Constituents de la Mat` eria, Universitat de Barcelona, Avda. Diagonal 647, E-08028 Barcelona, Spain Over more than fifty years, strangeness nuclear physics has disclosed may interesting phenomena related to the interactions of strange particles with nucleons. A brief overview of some developments in this field will be given, placing a special emphasis on the recent achievements in the physics of hypernuclei and on the possible existence of deeply bound kaonic states. Keywords: Hypernuclei; YN interaction; Kaonic nuclear states.

1. Introduction Nuclear physics with strangeness is a challenging field, where a variety of new and exotic phenomena can be investigated. The field emerged in the latest fifties, a few years after the discovery of the strange particles, such as the hyperons (Λ, Σ, . . . ) and kaons (K − , K + , . . . ). The early emulsion experiments already provided a considerable amount of data on the Λ binding energy in light hypernuclear species. However, a rapid evolution in understanding the behavior of hypernuclei was only realized in the seventies, thanks to the advent of counter experiments in accelerator laboratories such as CERN, Brookhaven National Laboratory and KEK, which disclosed new aspects of hypernuclear structure and produced many other hypernuclear species up to the very heavy ones. These new data allowed one to determine some aspects of the hyperon-nucleus interaction. In the eighties, thanks to the increase of the beam intensities, it became possible to measure the products from the decay of hypernuclei, opening a new direction of research, namely that of investigating the weak decay mechanism, which is still a not fully understood process. In the last fifteen years, the energy resolution of hypernuclear spectra, produced using the (π + , K + ) reaction at BNL and KEK, the (e, e0 , K + ) electromagnetic reaction at TJNAF or the stopped (K − , π) reaction with thin targets by the FINUDA collaboration at DaΦne, has been notably improved. Of especial relevance is the recent use of high resolution γ-ray detectors in the measurement of electromagnetic decays of hypernuclei, which has supplied a good amount of precise hypernuclear spectroscopic data, providing severe constraints on the hyperon-nucleon interaction. The latest hot topic in strangeness nuclear physics, which has raised great expectations

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over the last few years, has been the possible existence of very deeply bound kaonic nuclear state.s In this contribution, I will briefly overview some of the crucial discoveries in the field of strangeness nuclear physics. Special attention will be devoted to aspects of hypernuclear structure, to the recent developments in hypernuclear weak decay, and to the present status on the deeply bound kaonic systems.

2. Hypernuclear Structure A hypernucleus is a bound system of neutrons, protons and one or more strange baryons (hyperons), such as Λ, Σ, Ξ, . . . . The most common ones are Λ-hypernuclei, which are “stable” within the strong interaction, and decay via strangeness-changing weak processes with lifetimes of the order of 10−10 s. The structure of Λ hypernuclei has been reviewed recently.1 After the first discovery of a hypernuclear fragment in 1953,2 and for the following 20 years, more than 20 hypernuclear species were identified and the corresponding Λ binding energies were measured. Those data already disclosed some aspects of the hyperon mean field like a depth of only 2/3 of that of the nucleon. The recoilless-kinematics adopted in the pioneering counter-experiments at CERN in the seventies, using the exothermic (K − , π − ) reaction, transferred little momentum to the hyperon, hence producing hypernuclei very efficiently. The experiments were focussed on studying the structure of p-shell hypernuclei and a very small spin-orbit splitting was found.3 This can be explained within relativistic mean-field phenomenology 4 or, more recently, from the point of view of meson exchange models within effective-field-theory.5,6 The (π + , K + ) reaction, used subsequently at BNL and later at KEK, confirmed these results.7 Moreover, the high momentum transferred to the hyperon populated also deeper lying Λ single-particle orbits in heavier hypernuclei8–10 and disclosed the weaker fragmentation suffered by the Λ within a nucleus by the mere fact of being distinguishable from the nucleons. A quantitative understanding of hyperon-nucleon (YN) and hyperon-hyperon (YY) interactions is necessary to investigate new aspects and new forms of hadronic matter, such as the high-density nuclear matter inside neutron stars, where the appearance of hyperons plays a crucial role in determining their composition and their properties. However, YN scattering data are scarce because suitable hyperon beams are difficult to achieve due to the short hyperon lifetimes. In this situation, hypernuclear spectroscopy reveals as the most efficient tool to obtain valuable information on the YN and YY interactions. The earlier and more modern baryon-baryon potential models based on meson exchange rely on SU(3) flavor symmetry to extend the nucleon-nucleon (NN) potentials to the strangeness sector.11–13 More recently, YN and YY potential models have also been obtained in the framework of effective field theory.14 The ΛN interaction does not contain a one-pion exchange contribution because of isospin conservation. Therefore, an important piece of this interaction comes from the cou-

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pling to intermediate ΣN states via two pion exchange and demands the solution of a coupled channels problem, when applied to hypernuclei in the form of G-matrix effective interactions. In addition, the Σ-mediated three-body interaction might have a non-negligible effect on the fine structure of hypernuclei. Microscopic many-body calculations of hypernuclear properties from a G-matrix obtained using the existing bare interactions have been performed15–18 and an overall agreement for the binding energies of Λ hypernuclei as well as for the ΛΛ bond energy 19,20 is obtained. However, the spin dependence of the YN interactions varies strongly from model to model, giving rise to quite different predictions for spin doublets in the spectra of hypernuclei.18 The field has experienced an important turnover in the last years, thanks to the development of precision hypernuclear γ-ray spectroscopy 21 with unprecedented resolution of a few keV. As an example, we quote the measured energy separation of 43(5) keV resolved for the 3/2+ − 5/2+ doublet in 9Λ Be. The J¨ ulich and the 1989 version of the Nijmegen potential models obtained a four times larger splitting, while the modified potentials reduced the splitting to 80 keV.18 While still away from the experimental value, this is an example on how the experimental observation severely constraints the properties of the YN interactions. Another theoretical perspective to hypernuclear structure is given by the phenomenological models of the YN interaction, aiming at determining a few parameters governing the strength of the different operatorial pieces of the effective two-body force as accurately as possible using the precise γ-ray spectroscopic data.22,23 With the prospect of the new γ-ray experiments planned at BNL and KEK, the second generation of (e, e0 K + ) experiments at JLab with energy resolution of 0.3 − MeV, data from DaΦne obtained with the (Kstop , π − ) reaction on thin targets, and the future data from the high beam intensity at J-PARC, the field of hypernuclear spectroscopy is expected to continue its raising projection, providing in the next years a much better knowledge of the YN interactions.

3. Hypernuclear Weak Decay A free Λ hyperon decays mainly through the pionic mechanisms (Λ → pπ − , nπ 0 with pN ∼ 100 MeV/c) and has a lifetime of τΛ = 2.632−10 s. These decay channels are strongly Pauli blocked in the medium since the nucleons would be accessing already occupied states. With the mesonic channels being suppressed, medium to heavy hypernuclei decay mainly through the non-mesonic channels induced by one nucleon (Λn → nn and Λp → np with pN ∼ 400 MeV/c) or by more nucleons (ΛN N → N N N with pN ∼ 340 MeV/c, . . . ). Useful reviews on this subject can be found in.24–26 Hypernuclei may be considered as a powerful “laboratory” for unique investigations of the baryon-baryon strangeness-changing weak interactions. An old challenge of hypernuclear decay studies has been to find a satisfactory theoretical explanation of the large experimental values of the ratio between the neutron- and proton-induced non-mesonic decay rates, Γn /Γp ≡ Γ(Λn → nn)/Γ(Λp → np). On

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the theoretical side, the more important mechanism for non-mesonic decay is that mediated by the exchange of a pion. But, because of its strong tensor component, it supplies very small ratios, typically in the interval 0.05 ÷ 0.20, although it can reproduce the total non-mesonic decay rates observed for light and medium hypernuclei. Other interaction mechanisms have been studied extensively: i) the inclusion in the ΛN → nN transition potential of mesons heavier than the pion (also including the exchange of correlated or uncorrelated two-pions)27–31 as those displayed in Fig. 1; ii) the inclusion of interaction terms that explicitly violate the ∆I = 1/2 rule;32,33 iii) the inclusion of the two-body induced decay mechanism34–37 and iv) the description of the short range ΛN → nN transition in terms of quark degrees of freedom,38,39 which automatically introduces ∆I = 3/2 contributions. In particular, these calculations found a reduction of the proton-induced decay width due to the opposite sign of the tensor component of K-exchange with respect to the one for π-exchange. Very recently, the ΛN → nN interaction has been studied within an effective field theory framework.40 In spite of all the efforts, the theoretical calculations do not exceed 0.5 for the ratio Γn /Γp , substantially smaller than the experimental values of around 1, albeit with large error bars, obtained in the past from single nucleon spectra.41–44 N

N

N

N

N S

π,η,ρ,ω

N π

W N

(a)

Fig. 1.

N

N S

K,K*

N (∆)

∆ (N)

Λ

N

S

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π

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N

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Λ

π

∆ (N) W

N

(b)

N π

π

S N (∆)

π

S N

Weak decay mechanisms: one meson, uncorrelated and correlated two-pion exchange.

This unsatisfactory situation has experienced a drastic change in the last 5 years, thanks to the experimental feasibility of measuring two–nucleon coincidences,45–48 together with improved theoretical calculations49–51 that combined a one-mesonexchange model describing the weak decay with an intranuclear cascade code for the nucleon final state interactions.52 The correlation observables permit a cleaner extraction of Γn /Γp from data than single-nucleon observables. In Fig. 2 we compare the theoretical and experimental nucleon–nucleon coincidence spectra from the decay of 5Λ He, both for the opening angle and kinetic energy correlations. The complete calculation (dot-dashed line) is in perfect agreement with the measured spectra. Reanalysis of experimental data, taking into account the FSI effects, provides nowadays neutron-to-proton ratios of around 0.3-0.4 for s- and p-shell hypernuclei. These values are reproduced by the most complete theories, resolving in this way one on the more long-standing puzzles in the field of weak hypernuclear decay. Another problem, which regards an asymmetry in the non-mesonic weak decay of polarized hypernuclei, raised recently. The (π + , K + ) reaction produces hypernuclei with a substantial amount of polarization. Due to interference between parity

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Tn+Tp (MeV)

Fig. 2. Opening angle and total energy spectra of np pairs from the decay of Taken from [47] and [49].

5 He Λ

hypernuclei.

violating and parity conserving amplitudes in the subsequent weak decay process, the intensity of protons from the Λp → np decay emitted parallel or antiparallel to the polarization axis is different, which results in a non-zero value of the up-down asymmetry. The experimental weak decay asymmetry is small, close to zero, 53,54 while the theoretical models predict negative and large values.28,30,55–58 Table 1.

Intrinsic and observable decay asymmetries for various hypernuclei. 5 He Λ

OME (no FSI) FSI, Tpth = 50 MeV OME + 2π + 2π/σ (no FSI) FSI, Tpth = 50 MeV EXP53 EXP54

−0.590 −0.455 +0.041 +0.030 0.11 ± 0.08 ± 0.04 +0.08 0.07 ± 0.08−0.00

11 B Λ

12 C Λ

−0.809 −0.698 −0.554 −0.468 −0.181 −0.207 −0.173 −0.179 −0.20 ± 0.26 ± 0.04 +0.18 −0.16 ± 0.28−0.00

The fit to experimental data, performed within an effective field theory approach,40 predicted a dominating central, spin-and isospin-independent contact

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term. Prompted by this work, some models incorporated the phenomenological exchange of a scalar-isoscalar σ-meson,57,58 but they were unable to account for the experimental value of the asymmetry. The contributions of uncorrelated and correlated two-pion-exchange to the non-mesonic weak decay has also been studied in Refs.29,59–61 The model of Ref. 29 included the exchange of π, K, ω, as well as uncorrelated plus correlated two-pions (2π + 2π/σ). The total two-pion-exchange contribution to the decay rates turned out to be moderate but its effect on the asymmetry parameter was not evaluated. This was done in a recent work, 62 which considered all the mechanisms in Fig. 1. It was found that the two-pion-exchange mechanisms modify moderately the partial decay rates but, as seen in Table 1, they have a tremendous influence on the asymmetries, bringing them to values that are in perfect agreement with the recent experimental data. 4. Deeply Bound Kaonic States? Over the last years, the theoretical study of the interactions of antikaons with nuclei has received a lot of attention due to the important implications on the possible realization of interesting physics phenomena like a kaon condensate in neutron stars or deeply bound kaonic nuclear states. 4.1. Theoretical situation Phenomenological fits to kaonic atoms suggested that antikaons would feel strongly attractive potentials of the order of −200 MeV at the center of the nucleus. 63 However, a satisfactory theoretical understanding of the size of the antikaon optical po¯ scattering amplitude which tential demands it to be linked to the elementary KN is dominated by the presence of a resonance, the Λ(1405), located only 27 MeV below threshold. This makes the problem to be a highly non-perturbative one. In ¯ mesons with nucleons has been treated within recent years, the scattering of K ¯ amplitude must inthe context of chiral unitary methods.64 The in-medium KN corporate the effect of Pauli blocking, the self-consistent dressing of antikaons, 65–67 as well as the pion dressing in the πΣ channel.68,69 The strength of the potential was shown to be moderately attractive and of the order of 40–70 MeV at nuclear matter density, and it also provided a good reproduction of the kaonic atom data in light and medium heavy nuclei.70,71 We can then conclude that the SU(3) chiral unitary model with the inclusion of all the medium corrections on the baryons and the mesons is supported by kaonic atom data. ¯ With this behavior for the KN dynamics in nuclei having been established, 72–74 the theoretical claims on the possible existence of narrow deeply bound states in light nuclei, predicting A = 3 I = 0 and I = 1 states with binding energies of around 100 MeV or more, was surprising. A closer look at the model interaction and the many-body approximations employed in these works reveals serious deficiencies pointed out in Ref. 75 and in Ref. 76. Among those: 1) The theoretical potential in72–74 eliminates the direct coupling of πΣ → πΣ and πΛ → πΛ in contradiction

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with the results of chiral theory.64 2) Self-consistency, shown to be essential in regulating the size of attraction felt by the antikaons, was not implemented. 3) Finally, short range correlations, which would prevent reaching the high densities found, were not incorporated realistically. An important turnover in the understanding of deeply bound kaonic systems has been given by recent few body calculations applied to the K − pp system. Threebody coupled channel Faddeev equations are solved in Ref. 77, using separable ¯ N )I=1 two-body interactions fitted to scattering data. A I = 1/2 three-body K(N state was found with a binding energy in the range B ∼ 50 − 70 MeV and a width Γ ∼ 95 − 110 MeV. The recent variational calculations of Ref. 78 implement, as a major change with respect the previous works,72–74 realistic short-range NN correlations, which prevent from reaching the high central densities obtained before. Although still preliminary, the first results point at a binding energy for the K − pp system of around 50 MeV and a total width for this state larger than 100 MeV, the absorption K − N N → Y N channel, not considered in,77 contributing by about 20%. 4.2. Re-analysis of experimental evidence Motivated by the theoretical works predicting deeply bound kaonic states, experiment KEK-E471 used the 4 He(stopped K − , p) reaction and reported79 a structure in the proton momentum spectrum, which was denoted as the tribaryon S 0 (3115) with strangeness S = −1. If assigned to the formation of a (K − pnn) bound state, it would have a binding energy of 194 MeV. In Ref. 75 the structure was interpreted to be due to the two-body absorption mechanism, K − N N → Σp, which in the reaction K −4 He → Σ− pd would produce a well defined narrow peak at pp ∼ 480 MeV/c, as in the experiment, if the Fermi motion of the spectator d is ignored. We note that the KEK group reported recently 80 some deficiencies in correcting the proton efficiencies of their previous experiment, and the new proton spectrum from the reaction 4 He(stopped K − , p) no longer shows a peak, although a signal has been observed by the FINUDA collaboration from K − capture on 6 Li that also was interpreted as coming from the two-nucleon K − absorption mechanism.81 This might well be due to Fermi motion effects, which have been shown82 to produce a broadening of the proton peak of ∼ 70 MeV for K − absorption in 4 He and of only of ∼ 20 MeV if the absorption takes place in 6 Li. Another experiment of the FINUDA collaboration has measured the invariant mass distribution of Λp pairs.83 The spectrum shows a narrow peak at 2340 MeV, which corresponds to the same signal seen in the proton momentum spectrum, namely K − absorption by a two-nucleon pair leaving the daughter nucleus in its ground state. Another wider peak is also seen around 2260 MeV, which is interpreted +2 in Ref. 83 as being a K − pp bound state with BK − pp = 115+6 −5 (stat)−3 (syst) MeV +2 +14 and having a width of Γ = 67−11 (stat)−3 (syst) MeV. In a recent work84 it was shown that this peak is generated from the interactions of the Λ and nucleon, produced

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after K − absorption, with the residual nucleus. This excites the nucleus through secondary collision of the p or Λ after the K − pp absorption process, similarly as to what happens in (p, p0 ) collisions, generating the quasi-particle peak. The resulting invariant mass spectra requiring at least a secondary collision of the p(n) or Λ after the K − pp(np) absorption process in 12 C is compared to the FINUDA data in Fig. 3. The main bump at 2260 MeV in the Λp invariant mass spectra is reproduced by the calculations. Therefore, the experimental Λp invariant mass spectrum of the FINUDA collaboration83 is naturally explained as a consequence of final state interactions of the particles produced in nuclear K − absorption as they leave the nucleus. Together with the interpretation of the peak seen in the proton momentum spectrum79,81 as coming from a two-nucleon kaon absorption mechanism, where the rest of the nucleus acts as an spectator and the daughter nucleus remains in its ground state given in Refs. 75 and 81, it seems then clear that there is at present no experimental evidence for the existence of deeply bound kaonic states. PΛ>300 MeV and cos Θ < −0.8 1.2

−

K pp−−> Λ p K−pn−−> Λ n

dσ/dMpΛ [arb.units]

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1 0.8

K−pn and K−pp Mg.s. −−> g.s. 2Mp+MK

0.6 0.4 0.2 0

2100

2150

2200

M

pΛ

Fig. 3.

2250

2300

2350

[MeV]

Invariant mass of Λp distribution for K − absorption in

12 C.

Taken from [81] and [84].

Acknowledgments I warmly thank all colleagues that collaborated in obtaining some of the results presented here, in particular C. Chumillas, G. Garbarino, V. Magas, E. Oset, A. Parre˜ no and H. Toki. This work is partly supported by contracts BFM2003-00856 and FIS2005-03142 from MEC (Spain) and FEDER, the Generalitat de Catalunya contract 2005SGR-00343, and the E.U. FLAVIANET network contract MRTN-CT2006-035482. This research is part of the EU Integrated Infrastructure Initiative Hadron Physics Project under contract number RII3-CT-2004-506078. References 1. O. Hashimoto and H. Tamura, Prog. Part. Nucl. Phys. 57, 564 (2006). 2. M. Danysz and J. Pniewski, Phil. Mag. 44, 348 (1953). 3. W. Br¨ uckner et al., Phys. Lett. B 79, 157 (1978).

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MANY-BODY METHODS FOR NUCLEAR SYSTEMS AT SUBNUCLEAR DENSITIES ARMEN SEDRAKIAN Institute for Theoretical Physics, J. W. Goethe-Universit¨ at, D-60054 Frankfurt am Main, Germany ∗ E-mail: [email protected] JOHN W. CLARK Department of Physics, Washington University, St. Louis, Missouri 63130, USA ∗ E-mail: [email protected] This article provides a concise review of selected topics in the many-body physics of low density nuclear systems. The discussion includes the condensation of alpha particles in supernova envelopes, formation of three-body bound states and the BEC-BCS crossover in dilute nuclear matter, and neutrino production in S-wave paired superfluid neutron matter. Keywords: Nuclear matter; Bose condensations; BCS-BEC crossover; weak interactions.

1. Introduction The physics of matter at subnuclear densities ρ ∈ [1011 − 1014 ] g cm−3 is of great interest for the astrophysics of compact objects. The “hot” stage of evolution of matter, in which temperatures are in the range of tens of MeV, is associated with the dynamics of supernova explosions. Knowledge of the equation of state, composition, and weak-interaction processes are of prime importance for an understanding the mechanism of explosion, the formation of neutrino spectra at the neutrinosphere, and the elemental abundances of the low-density matter in the supernova winds that are prerequisite for the onset of r-process nucleosynthesis. Days to weeks after the supernova explosion subnuclear matter has become “cold”, with temperatures T < 0.1 MeV. Moreover, the properties of the subnuclear matter forming the crust of a neutron star are of fundamental importance for the entire spectrum of observable manifestations of pulsars, ranging for example from superfluid rotation dynamics to magnetic field evolution to neutrino cooling. Nuclear matter at subnuclear densities is a strongly correlated system in which the relevant degrees of freedom are well established and the interactions are constrained by experiment. The challenge lies in the many-body treatment of this system where macroscopic quantum phenomena such as Bose–Einstein condensation

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of deuterons and alpha particles exist as well as the BCS pairing in neutron matter. Our aim here is to describe some of the many-body methods for dealing with such correlated states of matter. We will pay less attention to the physical setting and implications of the results; the reader concerned with these issues is referred to the original literature cited among the references. 2. Bose–Einstein Condensation: A Lattice Monte-Carlo Perspective In this section we describe an approach to interacting Bose systems which is valid in the vicinity of the critical temperature of Bose–Einstein condensation (BEC). The method was put forward in the context of dilute gases interacting via repulsive two-body forces1,2 and has since been reformulated for a strongly correlated system interacting with attractive two-body and repulsive three-body forces.3 The method has been applied to Bose condensation of alpha particles in infinite matter. (Alternative studies are based on hypernetted-chain summations.4 ) Consider a uniform, non-relativistic system of identical bosons described by the Hamiltonian # " Z ~2 † 2 4 6 3 ∇ψ (x)∇ψ(x) − µ|ψ(x)| + g2 |ψ(x)| + g3 |ψ(x)| , H= d x 2m where m is the alpha-particle mass, µ is the chemical potential, and ψ is the boson field. Below, we shall implement lattice regularization. The theory defined by Eq. (1) can be mapped onto an effective scalar field theory within the finite-temperature Matsubara formalism. Consider the fields ψ and ψ † as periodic functions of the imaginary time τ ∈ [−β, β], where β = 1/T is the inverse temperature. Next, decompose the fields into discrete Fourier series ∞ ∞ X X iων τ ψ(x, ων ) = e ψ(x, τ ) = ψ0 (x) + eiων τ ψ(x, τ ), (1) ν=−∞

ν=−∞, ν6=0

where the Fourier frequencies ων are the bosonic Matsubara modes ων = 2πiνT (with ν taking integer values). The Matsubara Green’s function is given by GM (ων , x) = [iων − (2m)−1 ∇2 + µ]−1 . Here the chemical potential may include any contribution from the momentum- and energy-independent part of the self-energy; we also assume that any momentum and energy dependent parts are absorbed in the mass and the wave-function renormalizations, respectively. Since µ → 0 near T c , the characteristic scales of spatial variations of the Green’s function with non-zero Matsubara frequencies are l = (2mων )−1/2 , which are of the order of the thermal wave-length λ = (2π/mT )1/2 . The contribution of the non-zero Matsubara modes to the sum in Eq. (1) will be neglected since we are interested in scales L l, which are characterized only by the zero-frequency modes. In terms of new real scalar fields † φ1 and p φ2 defined via the relations ψ0 = η(φ1 + iφ2 ) and ψ0 = η(φ1 − iφ2 ), where 2 η = m/~ β, the continuum action of the theory is given by ( ) Z X 1 r u w 2 2 3 S (φ) = d3 x [∂ν φ(x)] + φ(x)2 − [φ(x)2] + [φ(x)2] , (2) 2 ν 2 4! 6!

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where φ2 = φ21 + φ22 , r = −2βµη 2 , u = 4!βg2 η 4 and w = 6!βg3 η 6 . The action (2) describes a classical O(2) symmetric scalar φ6 field theory in three spatial dimensions (3D). The positive sextic interaction guarantees that the energy is bound from below, which would not otherwise be the case because of the negative sign of the quartic term describing the attractive two-body interactions. The characteristic length scale of the theory is set by the parameter u, which has the dimension of inverse length; the dimensionless parameter of the lattice theory is uaL , where aL is the lattice spacing. The thermodynamic functions of the model are obtained from the partition function Z Z = [dφ(x)]exp [−S (φ)] . (3)

For example, the expectation value of the particle number density is given by nα = hψ ∗ ψi = (βV )−1 ∂lnZ/∂µ, where V is the volume. The continuum theory is now discretized on a lattice by replacing the integrations over spatial coordinates by a summation over lattice sites. The discretized version of the continuum action (2) is ) ( X X 2 3 φL (x)φL (x + aˆ ν )φL (x)2 + λ 1 + φL (x)2 λ + ζ φL (x)2 , −2κ SL (φ) = i

ν

the ν summation being carried out over unit vectors in three spatial directions (nearest neighbor summation). The hopping parameter κ and the two- and threebody coupling constants λ and ζ are related to the parameters of the continuum action through a2L r = (1 − 2λ)/κ − 6, λ = aL κ2 u/6, and ζ = wκ3 /90. The lattice and continuum fields are related by φL (x) = (2κ/aL )1/2 φ(x). The components of the spatial vector xν are discretized at integral multiples of the lattice spacing aL : xµ = 0, aL , . . . (Lν − 1)aL . For a simple cubic lattice in 3D with periodic boundary conditions imposed on the field variable, one has φL (x + aL L) = φL (x) (N.B. for a box of length L there are L3 (real) variables within the volume (LaL )3 ). In Ref. 3, the field configurations on the lattice were evolved using a combination of the heatbath and local Metropolis algorithms, executing 105 − 106 equilibration sweeps for lattices sizes from 83 to 643 . Once the field values on the lattice were determined, these were transformed into their counterparts in the continuum theory to obtain the statistical average value hHi of the Hamiltonian, i.e., the grand canonical (thermodynamical) potential Ω as a function of density. The critical temperature Tc for Bose–Einstein condensation can be obtained from the simulations.3 In practice one obtains the density n(T ), rather than T (n) directly, at constant fugacity z → 1. Working at small fugacity log z = −0.1, the density of the system at constant small chemical potential can be computed and the associated temperature is then identified with Tc . 3. From Pair Condensation to Three-body Bound States The few-body bound states are of interest in the “hot” stage of compact stars to the extent that they can provide a very efficient source of opacity for neutrinos propa-

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gating through matter. This situation can be compared to that of radiative transfer in ordinary stars, where the photoabsorption on the weakly-bound negative ion of hydrogen (H− ) largely determines the opacity. If the system is isospin-symmetric, the pairing occurs in the 3 S1 −3 D1 partial-wave channel. There is a two-body bound state in this channel in free space – the deuteron; hence the BCS to BEC crossover arises in this new context.5–7 Following up on the conjecture of Nozières and Schmitt-Rink,5 one may attempt to describe this crossover within mean-field BCS theory. The central numerical problem then reduces to solution of the gap equation Z dp0 p02 X 3SD1 ∆l0 (p0 ) ∆l (p) = − [1 − 2f (E(p0 ))] , (4) Vll0 (p, p0 ) p 2 0 )2 + D(p0 )2 (2π) 0 E(p l =0,2 where D2 (k) ≡ (3/8π)[∆20 (k) + ∆22 (k)] is the angle-averaged neutron-proton gap function, V 3SD1 (p, p0 ) is the interaction in the 3 S1 −3 D1 channel, E(p) is the quasiparticle spectrum, and f is the Fermi distribution function. The chemical potential is determined self-consistently from the gap equation (4) and the expression for the density. We now outline an algorithm for numerical solution of the gap equation, which can be applied to arbitrary potentials that are attractive at large separations. 8 The method is effective in dealing with the hard core (short-range repulsion) in nuclear potentials and could be useful for other systems featuring short-range repulsive interactions. The starting point is the gap equation with an ultraviolet momentum cutoff Λ ΛP , where ΛP is of the order of the natural (soft) cutoff of the potential. Successive iterations, which generate approximant ∆i to the gap function from approximant ∆(i−1) (i = 1, 2, . . .), are determined from Z Λ 0 02 ∆(i−1) (p0 , Λ) dp p (i) V 3SD1 (p, p0 ) p [1 − 2f (E(p))] . (5) ∆ (p, Λ) = (2π)3 E(p0 )2 + D(i−1) (p0 , Λ)2

The process is initialized by first solving Eq. (4) for D(pF ), where pF is the Fermi momentum, assuming the gap function to be a constant. The initial approximant for the momentum-dependent gap function is then taken as ∆(i=0) (p) = V (pF , p)D(pF ). Two iteration loops are implemented at given chemical potential. An internal loop operates at fixed Λ and solves the gap equation (5) iteratively for i = 1, 2, . . . . An external loop increments the cutoff Λ until d∆(p, Λ)/dΛ ≈ 0. The finite range of the potential guarantees that the external loop converges once the entire momentum range spanned by the potential is covered. Thus, choosing the starting Λ small enough, we execute the internal loop by inserting ∆(i−1) (p) in the right-hand side of Eq. (5) to obtain a new ∆i (p) on the left-hand side, which in turn is re-inserted in the right-hand side. This procedure converges rapidly to a momentum-dependent solution for the gap equation for ∆(p, Λj )θ(Λj − p), where θ is the step function and the integer j counts the iterations in the external loop. For the next iteration, the cutoff is incremented to Λj = Λj−1 + δΛ, where δΛ Λj , and the internal loop is iterated until convergence is reached. The two-

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loop procedure is continued until Λj > ΛP , after which the iteration is stopped, a final result for ∆(p) independent of the cutoff having been achieved. Once this process is complete, the chemical potential must be updated via the equation for the density. Accordingly, a third loop of iterations seeks convergence between the the output gap function and the chemical potential, such that the starting density is reproduced. The BCS-BEC crossover5 in nuclear systems has been studied as a function of density and asymmetry in the population of isospin states (proton-neutron asymmetry). A number of interesting features are revealed.7,9 In the extreme low-density limit, the chemical potential changes its sign and tends to −1.1 MeV, half the deuteron binding energy. Thus, twice the chemical potential plays the role of the eigenvalue in the Schr¨ odinger equation for two-body bound states in this, the BEC limit. The pair function is very broad in this low-density regime, indicating that the deuterons are well localized in space; conversely it is peaked in the BCS limit at high density where the Cooper pairs are correlated over large distances. In the case of asymmetric systems, the density distribution of the minority particles (protons) has a zero-occupation (blocking) region that is localized around their Fermi-surface. Upon crossover to the BEC side, the blocking region becomes wider and moves toward lower momenta. Eventually there is a topological change in the Fermi surface: the states are occupied starting at some finite momentum and are empty below that point. The Nozières–Schmitt-Rink conjecture5 of a smooth crossover from the BCS to the BEC limit does not hold in general for asymmetric systems; instead, phases with broken space symmetries intervene within a certain range of population asymmetries (see Ref. 9 and references therein). The three-body bound states can be computed from the three-body scattering matrix, which is written as T = T (1) + T (2) + T (3) , with components defined (using operator notation for compactness) as Q3 T (i) + T (j) . (6) T (k) = Tij + Tij Ω − 1 − 2 − 3 + iη

These are nonsingular type II Fredholm integral equations; the operator Q3 in momentum representation is given by Q3 (k1 , k2 , k3 ) = [1−f (k1 )][1−f (k2 )][1−f (k3 )]− f (k1 )f (k2 )f (k3 ); and i (ki ) are the quasiparticle spectra. The momentum space for the three-body problem is conveniently spanned by the Jacobi four-momenta K = ki + kj + kk , pij = (ki − kj )/2, and qk = (ki + kj )/3 − 2kk /3. The Tij -matrices are essentially the two-body scattering amplitudes, embedded in the Hilbert space of three-body states. In the momentum representation they are determined from Z 00 0 02 Q2 (p, q) dp p 0 0 hp0 |V |p00 i hp00 |T (ω)|p0 i, hp|T (ω)|p i = hp|V |p i + 4π2 ω − + (q, p) − − (q, p) + iη (7) where Q2 (q, p) = h1 − f (q/2 + p) − f (q/2 − p)i and ± (q, p) = h(q/2 ± p)i are averaged over the angle between the vectors q and p. Compared to the free-space problem, the three-body equations in the background medium now include two-

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Fig. 1. Dependence of the two-body (Ed ) and three-body (Et ) binding energies on inverse temperature, for fixed values of the ratio f = n0 /n, where n is the baryon density and n0 = 0.16 fm−3 is saturation density of nuclear matter. For asymptotically large temperature, E d (∞) = −2.23 MeV and Et (∞) = −7.53 MeV. The ratio Et (β)/Ed (β) is a universal constant independent of temperature.8

and three-body propagators that account for (i) the suppression of the phase-space available for scattering in intermediate two-body states, encoded in the functions Q2 , (ii) the phase-space occupation for the intermediate three-body states, encoded in the function Q3 , and (iii) renormalization of the single-particle energies (p). For small temperatures the quantum degeneracy is large and the first two factors significantly reduce the binding energy of a three-body bound state; at a critical temperature Tc3 corresponding to Et (β) = 0, the bound state enters the continuum. This behavior is illustrated in Fig. 1, which shows the temperature dependence of the two- and three-body bound-state energies in dilute nuclear matter for several values of the density of the environment. In analogy to the behavior of the inmedium three-body bound state, the binding energy of the two-body bound state enters the continuum at a critical temperature Tc2 , corresponding to the condition Ed (β) = 0. Our solutions exhibit a remarkable feature: the ratio η = Et (β)/Ed (β) is a constant independent of temperature. For the chosen potentials, the asymptotic free-space values of the binding energies are Et (0) = −7.53 MeV and Ed (0) = −2.23 MeV; hence η = 3.38. An alternative definition of the critical temperature for 0 trimer extinction is Et (β3c ) = Ed (β). This definition takes into account the breakup channel t → d + n of the three-body bound state into the two-body bound state d and a nucleon n. The difference between the two definitions is numerically insignificant. Fig. 2 depicts the normalized three-body bound-state wave function for three representative temperatures, as a function of the Jacobi momenta p and q. As the temperature drops, the wave function becomes increasingly localized around

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Fig. 2. Wave function of the three-body bound state as a function of the Jacobi momenta p and q defined in the text, for f = n0 /n = 60 and temperatures T = 60 (left panel) and 6.6 MeV (right panel).

the origin in momentum space. Correspondingly, the radius of the bound state increases in r-space, eventually tending to infinity at the transition. The wavefunction oscillates near the transition temperature (right panel of Fig. 2). This oscillatory behavior is a precursor of the transition to the continuum, which in the absence of a trimer-trimer interaction is characterized by plane-wave states. 4. The Weak Interaction in Cold Subnuclear Matter Non-nucleonic channels of cooling that operate in the crusts of neutron stars are electron neutrino bremsstrahlung off nuclei and plasmon decay:10 e + (A, Z) → e + (A, Z) + ν + ν¯ and plasmon → +ν + ν¯. Above the critical temperature T c for neutron superfluidity, the neutrons that occupy continuum states (i.e., those not bound in clusters) emit neutrinos of all flavors f via the bremsstrahlung process 11 n + n → n + n + νf + ν¯f . At T ≤ Tc the latter process is suppressed exponentially by exp(−2∆/T ), where ∆ is the gap in the quasiparticle spectrum. The superfluid nature of the matter allows for a neutrino-generating reaction (known as pair-breaking and recombination), whose rate scales like ∆7 and thus is specific to the superfluid (i.e., vanishes as ∆ → 0). The rate of the process is given by the polarization tensor of superfluid matter.12 A systematic diagrammatic method to compute the reaction rates is based on the kinetic equation for neutrino transport, formulated in terms of real-time Green’s functions.13 The corresponding Boltzmann equation is Z ∞ dq0 < [∂t + ∂q ων (q)∂x ] fν (q, x) Tr Ω (q, x)S0> (q, x) − Ω> (q, x)S0< (q, x) , 2π 0

where q ≡ (q0 , q) is the four momentum, S0>,< (q, x) are the neutrino propagators, and Ω>,< (q, x) are their self-energies. In second Born approximation with respect to the weak vertices ΓµL q1 , the latter are given in terms of the polarization tensor(s)

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Fig. 3. The sum of polarization tensors that contribute to the neutrino emission rate. The contributions form Π(b) (q) and Π(c) (q) vanish at the one-loop approximation.

Π>,< µλ (q1 , x) of ambient matter as Z d4q1 d4q2 >,< (2π)4 δ 4 (q−q2 −q1 )iΓµL q1 iS0< (q2 , x)iΓ†Lλq1 iΠ>,< −iΩ (q, x) = µλ (q1 , x), (2π)4 (2π)4 (8) The “greater” and “lesser” signs refer to the ordering of two-point functions along the Schwinger contour in the standard way. It is the total loss of energy in neutrinos per unit time and unit volume, i.e., the emissivity, that is of interest for the astrophysics of compact stars. This quantity is obtained by integrating the first moment of Boltzmann equation. For the bremsstrahlung of neutrinos and anti-neutrinos of given flavor it is expressed as Z G2 X d4 q δ(q1 + q2 − q)q0 g(q0 )Λµζ (q1 , q2 )Im Πµζ (q), (9) ν ν¯ = − 4 q ,q 1

2

where G is the weak coupling constant, q is the four-momentum transfer, g(q0 ) = [exp(q0 /T ) − 1]−1 is the Bose distribution function, Πµζ (q) is the retarded polarization tensor, and Λµλ (q1 , q2 ) = Tr [γ µ (1 − γ5) 6 q1 γ ν (1 − γ5) 6 q2 ]. Sums over the neutrino momenta q1,2 indicate integration over the invariant phase-space volume. The central problem of the theory is to compute the polarization tensor of the cold subnuclear matter. Initially, calculations of the polarization tensor within the superfluid phase were carried out at the one-loop approximation. This treatment was recently shown to be inadequate for the S-wave superfluid in neutron-star crusts.14 A many-body framework that is consistent with the sum rules for the polarization tensor, in particular with the f sum rule Z lim dω ω ImΠV (q, ω) = 0, (10) q→0

is provided by the random-phase resummation of the particle-hole diagrams in the superfluid matter. Because of the Nambu–Gorkov extension of the number of possible propagators in the superfluid phase, which now include both normal (G) and anomalous (F ) ones, at least three topologically different vertices are involved, which obey (schematically) the following equations ˆ 1 = Γ0 + v(GΓ1 G + Fˆ Γ3 G + GΓ2 Fˆ + Fˆ Γ4 Fˆ ), Γ ˆ2 = Γ v(GΓ2 G† + Fˆ Γ4 G† + GΓ1 Fˆ + Fˆ Γ3 Fˆ ), ˆ3 = Γ v(G† Γ3 G + Fˆ Γ1 G + G† Γ4 Fˆ + Fˆ Γ2 Fˆ ),

(11) (12) (13)

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kF [fm−1 ] 0.8 1.6

m∗ /m 0.97 0.84

∆ [MeV] 3.15 0.57

Tc [MeV] 1.78 0.38

R(0.5) 0.014 0.022

R(0.8) 1.0 1.0

R(0.9) 6.5 7.4

Note: Quoted are the wave number of neutrons, their effective mass, the gap and critical temperature, and the ratio R as a function of reduced temperature T /T c .

v being the scalar interaction in the particle-hole channel. The fourth integral equation for the vertex Γ4 follows upon interchanging particle and hole propagators in Eq. (11). The full polarization tensor is the sum of the contributions shown in Fig. 3. It can be expressed through “rotated” polarization functions A, B, and C as ΠV (q) =

A(q)C(q) + B(q)2 C(q) − v V [A(q)C(q) + B(q)2]

(14)

with A = 2∆2 I0 (q) − ∆2 ξq IA (q), B = −ω∆ I0 (q), and C(q) = −(ω 2 /2) I0 (q) + ξq IC (q), where ξq = q 2 /2m is the nucleon recoil. (The integrals I0 , IC , and IA can be found in Ref. 14.) It is now manifest that ΠV (q) = 0 when q = 0. Thus, the leading order contribution to the polarization tensor appears at O(q 2 ) and is linear in ξq . Since the neutrinos are thermal, with energies ω ∼ |q| ∼ T , the polarization tensor is suppressed by a factor T /m, which is of order 5 × 10−3 . The emissivity of the pair-breaking process can be compared to that of the modified bremsstrahlung (MB) process n + n → n + n + ν + ν¯, which is suppressed by roughly a factor exp(−2∆/T ) in the superfluid phase. Thus, the ratio of the neutrino loss rate through MB to that from the pair-breaking process, as computed by Friman and Maxwell,11 is 2 2 ∗ 4 Z ∞ x 2 F gA T 2∆ mn 2460π 4 κ exp − , , I0 = dxx5 f R= 2∆ 14175 cV ∆ mπ I0 T 2 T where gA and cV are the weak axial and vector coupling constants, m∗n and mπ are the neutron and pion masses, and F ' 0.6 (defined in Ref. 11). The factor κ = 0.2 accounts for the correction to the one-pion-exchange rate due to the full resummation of ladder series in neutron matter. The pair-breaking process dominates the MB process for temperatures below 0.8Tc , where it is most efficient. This is illustrated in the table above. The comparison made here should be taken with caution, since the exponential suppression of the MB rate is not accurate (within a factor of a few) at temperatures close to the critical temperature. Nevertheless, one may safely conclude that the vector-current pair-breaking process is competitive with the modified pair-bremsstrahlung process in the relevant temperature domain T /Tc ∈ [0.2 − 1]. 5. Closing Remarks Subnuclear matter at finite temperatures offers a fascinating arena for the development of many-body theory. Since the interactions invoved are well constrained by experiment, the entire complexity arises from the many-body correlations. As shown

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in the examples chosen, subnuclear matter may exhibit a range of salient manybody phenomena, such as Bose–Einstein condensation of alpha particles, BEC-BCS crossover in the deuteron channel, many-body extinction of bound states with increasing degeneracy, and non-trivial and quantitatively important modifications of the weak interaction rates due to many-body effects. Acknowledgments We thank H. M¨ uther and P. Schuck for their contribution to the research described in this article. We are grateful to the organizers of RPMBT14 for their impressive efforts and dedication in arranging a most successful conference. References 1. 2. 3. 4. 5. 6. 7. 8. 9.

10. 11.

12.

13. 14.

G. Baym, J.-P. Blaizot, and J. Zinn-Justin, Europhys. Lett. 49 (2), 150 (2000). J. Zinn-Justin, arXiv:hep-ph/0005272. A. Sedrakian, H. M¨ uther and P. Schuck, Nucl. Phys. A 766, 97-106 (2006). M. T. Johnson and J. W. Clark, Kinam 3, 3 (1980) also made available at this URL http://wuphys.wustl.edu/Fac/facDisplay.php?name=Clark.txt P. Nozières and S. Schmitt-Rink, J. Low Temp. Phys. 59, 195 (1985). T. Alm, B. L. Friman, G. R¨ opke and H. Schulz, Nucl. Phys. A 551, 45 (1993). U. Lombardo, P. Nozières, P. Schuck, H. J. Schulze and A. Sedrakian, Phys. Rev. C 64, 064314 (2001) [arXiv:nucl-th/0109024]. A. Sedrakian and J. W. Clark, Phys. Rev. C 73, 035803 (2006). A. Sedrakian and J. W. Clark, in ”Pairing in Fermionic Systems: Basic Concepts and Modern Applications”, eds. A. Sedrakian, J. W. Clark, and M. Alford, World Scientific, pp. 145-175, [arXiv:nucl-th/0607028]. G. G. Festa and M. A. Ruderman, Phys. Rev. 122, 1317 (1969); J. B. Adams, M. A. Ruderman, and C. H. Woo, Phys. Rev. 129, 1383 (1963). O. V. Maxwell and B. L. Friman, Astrophys. J. 232, 541 (1979); D. N. Voskresensky and A. V. Senatorov, Sov. J. Nucl. Phys. 45, 411 (1987) [Yad. Fiz. 45, 657 (1987)]; A. Sedrakian and A. E. L. Dieperink, Phys. Lett. B 463; E. van Dalen, A. E. L. Dieperink, and J. A. Tjon, Phys. Rev. C 67, 580 (2003). E. G. Flowers, M. Ruderman, and P. G. Sutherland, Astrophys. J. 205, 541 (1976); D. N. Voskresensky and A. V. Senatorov, Sov. J. Nucl. Phys. 45, 411 (1987) [Yad. Fiz. 45, 657 (1987)]. A. B. Migdal, E. E. Saperstein, M. A. Troitsky, and D. N. Voskresensky, Phys. Rep. 192, 179 (1990). A. Sedrakian and A. E. L. Dieperink, Phys. Rev. D 62, 083002 (2000); A. Sedrakian, arXiv:astro-ph/0701017; see also D. N. Voskresensky and A. V. Senatorov in Ref. 12. A. Sedrakian, H. M¨ uther and P. Schuck, arXiv:astro-ph/0611676; see also L. B. Leinson and A. Perez, Phys. Lett. B 638, 114 (2006) [arXiv:astro-ph/0606651].

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CORRELATIONS AS A FUNCTION OF NUCLEON ASYMMETRY: THE LURE OF DRIPLINE PHYSICS W. H. DICKHOFF Department of Physics, Washington University St. Louis, Missouri 63130, USA E-mail: [email protected] Experimental data describing elastic scattering of nucleons on Ca isotopes are employed to construct the nucleon self-energy at positive energies for these systems using a dispersive optical model analysis. Data below the proton Fermi energy, obtained from the (e, e0 p) reaction and the energies of low-lying single-particle orbits are employed to complete the determination of the nucleon self-energy in a broad energy domain that includes a few hundred MeV above and below the Fermi energy. The present analyis allows an extrapolation to larger nucleon asymmetry δ that demonstrates that protons should become more correlated with increasing δ. Keywords: Correlations in nuclei; spectroscopic factors; dispersive optical model; nucleon asymmetry; dripline physics.

1. Introduction It is generally known that elastic scattering of a fermion from a system comprised of the same species of fermions is related to the self-energy of such a fermion at positive energies.1 The most significant development in constructing the nucleon self-energy from the analysis of such data has been the dispersive optical model (DOM) approach.2 In this framework the emphasis is on the imaginary part of the self-energy which is responsible for the description of the loss of flux in the elastic scattering channel. Integrated properties and radial moments of this imaginary part are well constrained by the experimental data. The resulting imaginary part of the self-energy can then be used to determine the associated real part by employing the relevant dispersion relation. In addition to the scattering data Mahaux and Sartor have plausibly argued that the imaginary part associated with the coupling to surface excitations of the nucleus, is symmetric with respect to the Fermi energy. It is therefore also possible to describe the nucleon self-energy at energies below the Fermi energy. By utilizing a subtracted dispersion relation Mahaux and Sartor were able to incorporate the resulting energy-independent Hartree–Fock-like contribution as a local potential with a standard energy dependence that reflects well known nonlocality properties.3 While this analysis has been widely applied to individual nuclei, it becomes a tool with predictive power when data pertaining

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to more than one nucleus are included. Such a development was reported for the −Z closed-shell nuclei 40 Ca and 48 Ca.4 The nucleon asymmetry is given by δ= NA and the analysis of Ref. 4 yields the dependence of the proton self-energy on this asymmmetry parameter making it possible to extrapolate to systems with larger asymmetry such as they will be studied at radioactive beam facilities that are under construction or planned in the future. The present status of the understanding of nucleon correlations is briefly reviewed in Sec. 2. Some results of the DOM analysis of Calcium isotopes are illustrated in Sec. 3 while conclusions are drawn in Sec. 4. 2. Current understanding of Nucleon Correlations A full understanding of nuclear properties requires knowledge of the correlations between the nucleons. These correlations cause the spectroscopic strength of singleparticle levels to be reduced relative to independent-particle-model (IPM) values. Furthermore, the strength is spread in energy and, for stable closed-shell nuclei, the spectroscopic factors measured in (e,e 0 p) reactions are about 65% of the IPM predictions. The theoretical interpretation5 of these observations points to a global depletion of the shell-model Fermi sea due to short-range correlations (SRC) accompanied by a complementary presence of high-momentum components that have recently been observed.6 The depletion of the Fermi sea has also been studied experimentally with the (e, e0 p) reaction for protons in 208 Pb that puts the depletion of the proton Fermi sea at a little less than 20%7 in accordance with earlier nuclear matter calculations.8 The experimental occupation numbers of the various proton shells for this nucleus is strikingly similar to the one for nuclear matter and exhibits the effect of SRC as a global depletion, essentially independent of the nature of sp orbit. Superimposed on this global depletion is an enhanced depletion for those orbits that reside in the vicinity of the Fermi surface. The latter depletion is associated with the coupling of such particles with low-energy (mostly) surface excitations 9 and can therefore be regarded as due to long-range correlations. A quantitative understanding of the spectroscopic factors obtained from the (e,e 0 p) reaction therefore requires a substantial contribution from long-range correlations that represents the coupling of the single-particle states to low-lying collective excitations that are dominated by surface properties of the nucleus.5 3. Results and Extrapolations of the Dispersive Optical Model With an increasing interest in nuclei far from stability, it is important to understand how these correlations are modified as one approaches the drip lines. Nuclear-matter calculations suggest that protons (neutrons) feel stronger (weaker) correlations with increasing neutron fraction.10 These effects are related to the increased (decreased) importance of the stronger p-n tensor interaction compared to the n-n or p-p interactions for protons (neutrons) with increasing asymmetry. In addition to these volume effects associated with short-range and tensor correlations, one must con-

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Fig. 1. Comparison of experimental proton single-particle levels. For the levels indicated with the solid dots, their energies were included in the fits. The dashed lines indicate the Fermi energy.

sider the longer-range correlations associated with the coupling surface excitations that are present in finite nuclei. The asymmetry dependence of these latter correlations has not been well studied. Although there are numerous studies of the effect of correlations on the properties of sp levels for nuclei near stability, there are only a few studies for very neutron or proton-rich nuclei. From neutron knock-out reactions, Gade et al.11 infer that the removal strength of the 0d5/2 neutron hole state in the proton-rich 32 Ar nucleus is considerably reduced relative to that for stable nuclei. An appropriate method to study single-particle distributions is through the use of the dispersive optical model developed by Mahaux and Sartor.2 This description employs the Kramers-Kronig dispersion relation that links the imaginary and real parts of the nucleon self-energy.12 This procedure links optical-model (OM) analyses of reaction data at positive energies to structural information at negative energies. In Ref. 4, the properties of proton levels in Ca nuclei as a function of asymmetry have been investigated with the DOM. Previously measured elastic-scattering and reaction-cross-section data for protons on 40 Ca and 48 Ca as well as level properties of hole states in these nuclei, inferred from (e, e0 p) reactions, were simultaneously fit. The dependence on δ is extracted and used to predict level properties of 60 Ca. Results of this procedure yield good agreement for all considered scattering data and structure information in a large energy window.4,13 A fit to all these data requires a stronger imaginary surface potential near the Fermi energy leading to deeper binding of proton levels in 48 Ca as compared to 40 Ca. The effect on the level structure around the Fermi energy that is obtained for 60 Ca after extrapolating the potential

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is shown in Fig. 1. The levels in the immediate vicinity of EF are focused closer to EF , increasing the density of sp levels. A reduced gap between the particle and hole valence levels implies that the closed-shell nature of this nucleus has diminished and proton pairing may be important. This possibility of proton pairing at large nucleon asymmetry therefore has as its origin the larger strength of p-n relative to the p-p and n-n interactions. Hence, protons (neutrons) experience larger (weaker) correlations in neutron-rich matter. The reversed is true for proton-rich matter. 4. Conclusion The exciting prospect of studying nuclei at large asymmetry in future radioactive beam facilities requires the development of tools to study the relevant scientific questions. The presented framework that relies on the Green’s function approach in its DOM incarnation provides such a vehicle. It can identify missing data that allow better extrapolations and predictions of properties of nuclei towards the dripline and can be continuously updated and refined to take new data into account. The main conclusion at present is that an increase of the correlations of one kind of particle is to be expected when a large excess of the other kind is present. Acknowledgments The work reported in this contribution is based on research performed in collaboration with Bob Charity, Jon Mueller and Lee Sobotka at Washington University in St. Louis. This work is supported by the U.S. National Science Foundation under Grant No. PHY-0652900 and the U.S. Department of Energy, Division of Nuclear Physics under Grant No. DE-FG02-87ER-40316. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

J. S. Bell and E. J. Squires, Phys. Rev. Lett. 3, 96 (1959). C. Mahaux and R. Sartor, Adv. Nucl. Phys. 20, 1 (1991). F. Perey and B. Buck, Nucl. Phys. 32, 353 (1962). R. Charity, L. G. Sobotka and W. H. Dickhoff, Phys. Rev. Lett. 97, 162503 (2006). W. H. Dickhoff and C. Barbieri, Prog. Part. Nucl. Phys. 52, 377 (2004). D. Rohe et al., Phys. Rev. Lett. 93, 182501 (2004). M. F. van Batenburg, Deeply bound protons in 208 Pb, PhD thesis, University of Utrecht (2001). B. E. Vonderfecht, W. H. Dickhoff, A. Polls and A. Ramos, Phys. Rev. C 44, R1265 (1991). M. G. E. Brand, G. A. Rijsdijk, F. A. Muller, K. Allaart and W. H. Dickhoff, Nucl. Phys. A 531, 253 (1991). T. Frick, H. M¨ uther, A. Rios, A. Polls and A. Ramos, Phys. Rev. C 71, 014313 (2005). A. Gade et al., Phys. Rev. Lett. 93, 042501 (2004). W. H. Dickhoff and D. Van Neck, Many-Body Theory Exposed! (World Scientific, New Jersey, 2005). R. Charity, J. Mueller, L. G. Sobotka and W. H. Dickhoff, Phys. Rev. C (2007), to be published.

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FERMI HYPERNETTED CHAIN DESCRIPTION OF DOUBLY CLOSED SHELL NUCLEI F. ARIAS DE SAAVEDRA1 , C. BISCONTI2 , AND G. CO’2 1 Departamento

de F´ısica At´ omica, Molecular y Nuclear, Universidad de Granada, E-18071 Granada, Spain E-mail: [email protected] 2 Dipartimento di Fisica, Universit` a del Salento, and INFN sezione di Lecce, I-73100 Lecce, Italy For the first time, Fermi HyperNetted Chain (FHNC) techniques have been applied to describe the ground-state of medium and heavy doubly closed shell nuclei, with fully realistic nuclear interactions, including both two- and three-body forces, and operator dependent correlation functions. Calculations for the 12 C, 16 O,40 Ca, 48 Ca and 208 Pb nuclei, have been done by using Argonne V8’ two-body potential together with Urbana IX three-body force. These calculations reach an accuracy comparable to that of the best nuclear matter variational calculations. We have also investigated the effects produced by the short range correlations (SRC) on some ground state quantities related to observables. Keywords: Nuclear structure; Many-body theories; Closed shell nuclei.

1. Introduction The validity of the non relativistic description of the atomic nuclei has been well established in the last ten years.1 The idea is to describe the nucleus with a Hamiltonian of the type: H =−

A A X ~2 X 2 vij + ∇i + 2m i=1 j>i=1

A X

vijk ,

(1)

k>j>i=1

where the two- and three-body interactions, vij and vijk respectively, are fixed to reproduce the properties of the two- and three-body nuclear systems. In our case we have used the Argonne V8’ two-body potential together with Urbana IX three-body force.1 About fifteen years ago, we started a project aimed to apply to the description of medium and heavy nuclei the Fermi HyperNetted Chain (FHNC) techniques, successfully used to describe infinite systems of fermions. We solve the many-body Schr¨ odinger equation by using the variational principle within a subspace of the full Hilbert space spanned by the A-body wave functions which can be expressed as: Ψ(1, ..., A) = F(1, ..., A)Φ(1, ..., A) ,

(2)

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where F(1, ..., A) is a many-body correlation operator which considers the short range correlation (SRC). We indicate with Φ(1, ..., A) a Slater determinant composed by single particle wave functions, φα (i) that are eigenfunctions of the total angular momentum of the nucleon and may be different for protons and neutrons. The complexity of the nuclear interaction requires the use of an operator dependent correlation: 

F(1, ..., A) = S 

A Y

j>i=1



Fij  = S

 " 6 A  Y X 

j>i=1

p fp (rij )Oij

p=1

#  

(3)

where S is a symmetrizer operator and in the above equation we have adopted the nomenclature commonly used in this field, by defining the operators as: p=1,6 Oij = [1, σi · σ j , Sij ] ⊗ [1, τ i · τ j ] ,

(4)

where Sij indicates the tensor operator. The use of operator dependent correlations requires the Single Operator Chain (SOC) approximation to construct the Fermi Hypernetted Chain (FHNC) integral equations. The validity of this approximation is monitored by controlling the exhaustion of the sum rules for the one- and two-body distribution functions.2 In the following sections, we shall present some results obtained in the FHNC/SOC computational scheme applied to various doubly-closed shell nuclei.3 A more complete presentation of the FHNC techniques can be found in Ref. 4.

2. Ground State Energies In Table 1, we present the binding energy per nucleon of various doubly closed shell nuclei obtained by minimizing their energy functionals. We have indicated with T the kinetic energy, with V2−body the contribution of the two-body interaction, with VCoul the contribution of the Coulomb interaction and with V3−body the total contribution of the three-body force. The rows labeled T + V2 show the energies obtained by considering the two-body interactions only. We can see that the binding is provided by a subtle subtraction between the kinetic energy and the two-body potential energy. The three-body force provides a repulsive contribution as in the case of nuclear matter. These results are in contrast with those obtained in light nuclei with Monte Carlo techniques, where three-body forces always produce attractive contributions. Work is in progress to understand these differences. The comparison with the experimental energies indicates a general underbinding of about 4.0 MeV per nucleon. This is roughly the same underbinding obtained, at the saturation density, by the most recent FHNC/SOC nuclear matter calculations. 5 The behavior of the 12 C nucleus is anomalous in this general trend.

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154 Table 1. Energies per nucleon in MeV, obtained by using the AV 80 + U IX interaction. AV8’+UIX T V2−body VCoul T + V2 V3−body E Eexp

12 C 27.13 −29.38 0.67 −1.58 0.67 −0.91

−7.68

16 O 32.33 −38.63 0.86 −5.34 0.86 −4.48

−7.97

40 Ca 41.06 −49.36 1.97 −6.34 1.76 −4.58

−8.55

48 Ca 39.64 −46.95 1.57 −5.74 1.61 −4.14

−8.66

208 Pb

39.56 −48.88 3.97 −5.35 1.91 −3.43 −7.86

3. Other Observables We have studied the effects of the SRC on other ground state quantities: density, charge and momentum distributions, natural orbits, occupation numbers, quasiparticle wave functions and spectroscopic factors.2–4 All these quantities show a more or less pronounced sensitivity to the SRC. The momentum distributions are the quantities where the effects of the SRC are more evident. We show in Fig. 1 two representative examples of our results, relative to the 48 Ca and 208 Pb nuclei. Correlated and Independent Particle Model (IPM) distributions almost coincide in the low momentum region up to a precise value of k, when they start to deviate. The correlated distributions have high momentum tails, which are orders of magnitudes larger than the IPM results. In Fig. 1 the thicker lines show the results of our FHNC/SOC calculations, while the thinner ones those obtained in the Independent Particle Model. The results shown in Fig. 1 indicates that the differences between protons and neutrons momentum distributions are more related to the different single particle structures than to the correlation effects. The main differences in the two distributions is in the zone where the n(k) values drops of orders of magnitudes. This corresponds to the discontinuity region of the momentum distribution in the infinite systems, which is related to the Fermi momentum. In finite systems, the larger number of neutrons implies that the neutron Fermi energy is larger than that of the protons, and, consequently, the effective Fermi momentum. For this reason, the neutron momentum distributions drop at larger values of k than do the proton distributions. While the momentum distributions in the low k region is dominated by the IPM dynamics, in the higher k region the correlation plays an important role. In the low k region the proton and neutron momentum distributions have slightly different shapes, they are rather similar in the high k region. This indicates that the effects of the SRC are essentially the same for both kinds of nucleons. Our results are in agreement with the findings of Ref. 6, where the momentum distribution of asymmetric nuclear matter is presented. There is however a disagreement with the results of Ref. 7, where, always in asymmetric nuclear matter, correlations effects among protons were found to be stronger than those among neutrons.

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protons neutrons

10

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-3

Pb

-3

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10

48

-4

-4

Ca

10

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-5

10-5

10

10-6

-6

10 0.0

1.0

2.0

k [fm-1]

3.0

4.0

0.0

1.0

2.0

3.0

4.0

k [fm-1]

Protons (full lines) and neutrons (dashed lines) momentum distributions of the Ca and 208 Pb. The thick lines show the results of our calculations, the thin lines the IPM results.

Fig. 1.

48

4. Conclusion We have shown that FHNC/SOC calculations in doubly closed shell nuclei have reached the same degrees of accuracy than those done for nuclear and neutron matter. These calculations allows us to study effects that mean-field based effective theories cannot study. We have presented here the case of the momentum distributions, but we found other examples of the relevance of the SRC in other ground-state quantities.2–4 Acknowledgments This work has been partially supported by the agreement INFN-CICYT, by the Spanish Ministerio de Educaci´ on y Ciencia (FIS2005-02145) and by the MURST through the PRIN: Teoria della struttura dei nuclei e della materia nucleare. References 1. B. S. Pudliner, V. R. Pandharipande, J. Carlson, S. C. Pieper and R. B. Wiringa, Phys. Rev. C 56, p. 1720 (1997). 2. C. Bisconti, F. A. de Saavedra, G. Co’ and A. Fabrocini, Phys. Rev. C 73, p. 054304 (2006). 3. C. Bisconti, F. A. de Saavedra and G. Co’, Phys. Rev. C 75, p. 054302 (2007). 4. F. A. de Saavedra, C. Bisconti, G. Co’ and A. Fabrocini, Phys. Rep. (2007), in press; (nucl-th) arXiv:0706.3792 5. A. Akmal, V. R. Pandharipande and D. G. Ravenhall, Phys. Rev. C 58, p. 1804 (1998). 6. P. Bo˙zek, Phys. Lett. B 586, p. 239 (2004). 7. T. Frick, H. M¨ uther, A. Rios, A. Polls and A. Ramos, Phys. Rev. C 71, p. 014313 (2005).

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MANY-BODY CHALLENGES IN NUCLEAR-ASTROPHYSICS G. MARTÍNEZ-PINEDO Gesellschaft f¨ ur Schwerionenforschung, Planckstrasse 1, D-64291 Darmstadt, Germany Nuclear astrophysics combines inputs from different fields with the objective of explaining the abundances and evolution of chemical elements in the Universe. Future radioactive ion beam facilities will provide access to many of the nuclei that participate in explosive nucleosynthesis scenarios. To fully exploit the potential of these facilities progress in theoretical nuclear physics will be required. This article reviews different many-body methods currently used for the description of processes relevant for nuclear astrophysics.

1. Introduction Nuclear astrophysics aims to describe the nuclear processes responsible for energy generation in the universe and the synthesis of elements. This is achieved by a combination of observational astronomy, astrophysical modeling and experimental and theoretical nuclear physics inputs. In recent years a tremendous progress has been achieved in our observational capabilities in all the wave lengths of the electromagnetic spectrum and including the detection of neutrinos emitted from several astrophysical sources. This has allowed to obtain detailed abundances of elements for many stars with different metallicities including very metal poor stars. This observations challenge our current understanding of the synthesis of elements in stars as laid down 50 years ago by the seminar works of Burbidge, Burbidge, Fowler and Hoyle1 and Cameron2 and allows us to understand how the individual nucleosynthesis processes operate in single events and trace the nucleosynthesis history of the galaxy. The astrophysical modelling of the different events and in particular explosive scenarios like X-ray burst and supernovae explosions (both core-collapse and thermonuclear) is currently under development with the goal of achieving threedimensional simulations that include all the relevant physics. Experimental radioactive beam (RIB) programs are presently being pursued world-wide by major research laboratories. Upgrades to improve RIB intensities and R&D programs for the development of new facilities have started, providing research opportunities with nuclei closer to the driplines. The upgraded radioactive beam factory (RIBF) at RIKEN has already delivered the first beams. SPIRAL 2 at GANIL is planned to start operation in 2010. At GSI the future Facility for Antiproton and Ion Research (FAIR) is planned to deliver beams in 2012 opening

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an unprecedented range of experimental opportunities around a superconducting double-ring synchrotron, a system of storage rings for beam collection and cooling and a new superconducting fragment separator Super-FRS. All this facilities will allow us to obtain data of incomparable quality to be used in nucleosynthesis calculations. However, due to the fact that the astrophysical processes occur at finite temperatures that involve the contribution of excited states that are not always accessible experimentally and that in some cases like the r-process many of the relevant nuclei will still not be accessible by future facilities we need to develop and improve current theoretical models to fully exploit the nuclear astrophysics potential of this facilities.

2. Reactions Involving Light Nuclei In recent years ab-initio models have been developed for the description of the structure of light nuclei. The Green’s Function Monte-Carlo method3,4 has been very successful in the description of light nuclei with A ≤ 12 using the phenomenological high precision Argonne v18 two-nucleon potential supplemented by three-nucleon potentials. The ab-initio no-core shell model (NCSM)5,6 is a method to solve the nuclear many body problem for light nuclei using realistic inter-nucleon forces. The calculations are performed using a large but finite harmonic-oscillator (HO) basis. Due to the basis truncation, it is necessary to derive an effective interaction from the underlying inter-nucleon interaction that is appropriate for the basis size employed. The effective interaction contains, in general, up to A-body components even if the underlying interaction had, e.g. only two-body terms. In practice, the effective interaction is derived in a sub-cluster approximation retaining just two- or threebody terms. A crucial feature of the method is that it converges to the exact solution when the basis size increases and/or the effective interaction clustering increases. The method allows for the use of different realistic potentials like the Argonne v18 or the CD-Bonn potentials or even potentials obtained from chiral effective interaction theory.7 First attempts to extend this ab-initio models for the description of reactions relevant for nuclear astrophysics has been achieved recently. The variational Monte Carlo method has been applied by Nollet et al.8,9 to describe 2 H(α, γ)6 Li, 3 H(α, γ)7 Li, and 3 He(α, γ)7 Be capture reactions, while the NCSM has been applied by Navr´ atil et al.10,11 to the description of 7 Be(p, γ)8 B. Both models assume a potential model for the description of the scattering states but derive the important spectroscopic information of the states involved from a full ab-initio treatment using realistic interaction. In the NCSM calculations the overlap integrals have to be corrected due to the incorrect asymptotic properties of the used harmonic oscillator basics. This is achieved using a Wood-Saxon potential solution that matches the interior of the NCSM overlap integral or by a direct match with the Whittaker function. Light nuclei are characterized by the existence of states with a clear cluster

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structure. The Hoyle state in 12 C at a excitation energy of 7.654 MeV is probably the most famous one as it determines the fusion rate for three α particles. These cluster states are poorly described by shell-model type configurations like the ones used in the NCSM. The nuclear cluster model gives a unique description of nuclear bound and scattering states taking the Pauli principle among all nucleons fully into account.12 However, it is based on the assumption that the full many-nucleon wave function can be approximated by an antisymmetrized cluster product state where the internal degrees of freedom of the clusters are frozen and that the nuclear low-energy phenomena are solely determined by the dynamics of the relative motion among the clusters, which is governed by some (often effective) Hamiltonian. Nevertheless due to the potential selection of a basis made of multi-cluster wave functions supplemented by shell-model-like states, the cluster model has a large flexibility and can be the method of choice to describe several structure phenomena in light nuclei. Another strong point of the method is the ability to consistently describe bound, resonant and scattering states based on the same microscopic Hamiltonian. This makes the cluster model a very useful tool for the study of astrophysically important reactions between (relatively) light nuclei where a direct measurement of the cross sections at the astrophysically most effective energies is often impossible and the required information is achieved by extrapolation of data to lower energies. In an exciting recent development cluster model applications have been improved in two important aspects: (i) by introducing more flexible wave functions and (ii) by using more realistic NN interactions. These improved models (Antisymmetrized Molecular Dynamics (AMD)13 and Fermionic Molecular Dynamics (FMD)14 have been quite successfully applied to nuclear structure problems in light nuclei and the FMD has very recently been used to study astrophysically important reactions. In the FMD the many-body states are given by Slater determinants with Gaussian wave packets for the spatial degrees of freedom of the single-particle states. The intrinsic states of the cluster nuclei are determined by minimizing the intrinsic energy expectation value with respect to all the single-particle parameters (e.g. the complex width parameters of the Gaussians). The translational, rotational and parity symmetries of the intrinsic states are guaranteed by appropriate projections. Improved intrinsic wave functions are achieved by multi-configuration mixing where the basis configurations are obtained by minimizing the energy under constraints on collective variables like dipole, quadrupole or octupole moments. The important short-ranged nucleon-nucleon correlations are accounted for by using the Unitary Correlation Operator Method (UCOM);15,16 i.e. they are introduced by an unitary operator which is given by a product of a central and a tensor correlator. The parameters in these central and tensor correlators are determined by variation in the various spin-isospin channels of the two-nucleon system. The matrix elements of the momentum-dependent interaction, defined this way, are very similar to those of the Vlow−k in momentum space. No three-body interaction is taken explicitly into account, but it is simulated by a correction term which is fitted to the binding energies and radii of the double-magic nuclei 4 He, 16 O, and 40 Ca. Detailed

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descriptions of the FMD and its applications to nuclear structure are given in.14,17 The FMD is a very promising tool to describe astrophysically important nuclear reactions among light nuclei as it combines the flexibility in the choice of basis wave functions for bound and scattering states with the virtue to account for the relevant degrees of freedom and correlations among the nucleons. For example, the FMD reproduces the spectrum and the low-energy 3 He+ 4 He scattering phase shifts quite well.18 Hence it should be extended to a calculation of the 3 He(α, γ)7 Be cross section at solar energies; such a study is in progress. 35

S-factor [MeV barn]

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10 34 10 33 10 32 10 31 10 30 10 29 10 28 10 27 10 26 10 25 10 24 10 23 10 22 10 21 10 20 10 19 10 0

16

16

O- O 22 O- O 24 24 O- O 22

5

10

15

20

Ecm [MeV] Fig. 1.

FMD S-factors for the fusion reactions of various oxygen isotopes. 19

Another example for the capabilities of the FMD model is given in Fig. 1 which shows the astrophysical S-factors for the sub-barrier fusion of various oxygen isotopes.19 Fusion of two 16 O nuclei triggers oxygen burning as one of the last stages in stellar hydrostatic evolution, while the fusion of the neutron-rich oxygen isotopes 22 O and 24 O explores the potential increase of the fusion cross sections due to the pronounced neutron tails which develop in neutron-rich nuclei. The fusion of such neutron-rich isotopes is expected to be relevant for the evolution of the crust matter of a neutron star if the latter accumulates matter from a binary star and undergoes regular X-ray bursts.20 Two facts are worth mentioning from Fig. 1: At first, the FMD calculation reproduces the 16 O +16 O fusion data quite well, without adjustment of any parameters. Secondly, the pronounced neutron tail enhances the fusion cross sections for the other oxygen isotopes by several orders of magnitude stressing the sensitivity of the fusion process to a correct description of the asymptotic

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cluster wave functions. While FMD studies for the many pycnonuclear fusion reactions needed to simulate the neutron star crust evolution are yet computationally not feasible, such studies are, however, very useful to check and constrain simple phenomenological potential models.21 For medium and heavy nuclei where the density of states is high enough and many resonances contribute to the capture cross section the most frequently used model is the Hauser–Feshbach approach.22 For nuclei near the drip-lines or near closed shell configurations, the density of levels is not high enough for the Hauser– Feshbach approach to be applicable. For these cases alternative theoretical approaches such as the Shell Model Embedded in the Continuum23 or the Gamow Shell Model.24 At present only the traditional shell model has been applied to determine the relevant spectroscopic factors together with a potential model for the calculation of the scattering states. In this way, proton capture reaction rates for sd-shell25 and pf -shell nuclei26 necessary for rp process studies have been determined.

3. Weak Processes in Medium and Heavy Nuclei Nuclear beta-decay and electron capture are important during the late stages of stellar evolution (see Ref. 27 for a recent review). At the relevant conditions in the star electron capture and β decay are dominated by Gamow–Teller (and Fermi) transitions. Earlier determinations of the appropriate weak interaction rates were based in the phenomenological work of Fuller, Fowler and Newman.28–31 The shell model makes it possible to refine these estimates. For the sd shell nuclei, important in stellar oxygen and silicon burning, the relevant rates where determined in Ref. 32. More recently, it has been possible to extend these studies to pf shell nuclei relevant for the pre-supernova evolution and collapse33–35 using a G-matrix interaction based in a realistic potential with some minor phenomenological corrections to the monopoles of the interaction.36 The astrophysical impact of the shell-model based weak interaction rates have been recently studied by37,38 The basic ingredient in the calculation of the different weak interaction rates is the Gamow–Teller strength distribution. The GT+ sector directly determines the electron capture rate and also contributes to the beta-decay rate through the thermal population of excited states.29 The GT− strength contributes to the determination of the β-decay rate. To be applicable to calculating stellar weak interaction rates the shell-model calculations should reproduce the available GT+ (measured by (n, p)-type reactions) and GT− (measured in (p, n)-type reactions). Recently developed techniques, involving (d, 2 He) charge-exchange reactions at intermediate energies,39 have improved the energy resolution by an order of magnitude or more as compared with the (n, p)-type reactions. Figure 2 compares the shell-model GT + distribution computed using the KB3G interaction40 with a recent experimental measurement of the 51 V(d, 2 He) performed at KVI.41 Shell-model diagonalization techniques have been used to determine astrophys-

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0.4 51V(d,2He)51Ti

0.3 0.2 0.1

0.1 0.2

B(GT+)

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1

2

3

4

5

6 7 Ex [MeV]

Fig. 2. Comparison of the shell-model GT+ distribution (lower panel) for 51 V with the high resolution (d, 2 He) data.41 The shell-model distribution includes a quenching factor of (0.74)2 .

ically relevant weak interaction rates for nuclei with A ≤ 65. Nuclei with higher masses are relevant to study the collapse phase of core-collapse supernovae. 27 The calculation of the relevant electron-capture rates is currently beyond the possibilities of shell-model diagonalization calculations due to the enormous dimensions of the valence space. However, this dimensionality problem does not apply to Shell-Model Monte-Carlo methods.42 Moreover, the high temperatures present in the astrophysical environment makes necessary a finite temperature treatment of the nucleus, this makes SMMC methods the natural choice for this type of calculations. Initial studies of Ref. 43 showed that the combined effect of nuclear correlations and finite temperature was rather efficient in unblocking Gamow–Teller transitions on neutron rich germanium isotopes. More recently this calculations have been extended to cover all the relevant nuclei in the range A = 65–112.44 The resulting electron-capture rates have a very strong influence in the collapse44 and post-bounce45 dynamics. Knowledge of neutrino nucleus reactions is necessary for many applications,

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e.g. the neutrino oscillation studies, detection of supernova neutrinos, description of the neutrino transport in supernovae and nucleosynthesis studies. Most of the relevant neutrino reactions have not been studied experimentally so far and their cross sections are typically based on nuclear theory (see Ref. 46 for a recent review). The model of choice for the theoretical description of neutrino reactions depends of the energy of the neutrinos that participate in the reaction. For low neutrino energies, comparable to the nuclear excitation energy, neutrinonucleus reactions are very sensitive to the appropriate description of the nuclear response that is very sensitive to correlations among nucleons. The model of choice is then the nuclear shell-model. 0~ω calculations have been used for the calculation of neutrino absorption cross sections47 and scattering cross sections48 for selected pf shell nuclei relevant for supernovae evolution. For lighter nuclei complete diagonalizations can be performed in larger model spaces, e.g. 4~ω calculations for 16 49,50 O and 6~ω calculations for 12 C.51,52 Other examples of shell-model calculations of neutrino cross sections are the neutrino absorption cross sections on 40 Ar of53 for solar neutrinos (see Ref. 54 for an experimental evaluation of the same cross section), this cross section have been recently evaluated by46 for supernova neutrinos. And the evaluation by55 of the solar neutrino absorption cross section on 71 Ga relevant for the GALLEX and SAGE solar neutrino experiments. For higher neutrino energies the standard method of choice is the random phase approximation as the neutrino reactions are sensitive mainly to the total strength and energy centroids of the different multipoles contributing to the cross section. In some selected cases, the Fermi and Gamow–Teller contribution to the cross section could be determined from shell-model calculation that is supplemented by RPA calculations for higher multipoles. This type mix calculation has been carried out for several iron isotopes56,57 and for 20 Ne.58 Recently inelastic neutrino-nucleus scattering has been included for the first time in supernova simulations. The relevant cross sections have been calculated based on large-scale shell model calculations for the allowed GT transitions and within the random phase approximation for forbidden transitions,59 taking special care of finite temperature effects. At low and modest neutrino energies Eν the cross sections are dominated by GT0 contributions for which the shell model has been validated by detailed comparison to precision M 1 data derived from electron scattering on spherical nuclei which are mainly due to the same isovector response.60 Although inelastic neutrino-nucleus scattering contributes to the thermalization of neutrinos with the core matter, the inclusion of this process has no significant effect on the collapse trajectories. However, it increases noticeably the opacity for high-energy neutrinos after the bounce.61 As these neutrinos interact with the nuclei, they are down-scattered in energy, in this way significantly reducing the highenergy tail of the spectrum of emitted supernova neutrinos. This makes the detection of supernova neutrinos by earthbound detectors more difficult, as the neutrino detection cross section scales with Eν2 .62

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4. Nucleosynthesis of Heavy Nuclei The rapid neutron-capture process (r-process) is responsible for the synthesis of approximately half of the nuclei in nature beyond Fe.1,2,63 It requires neutron densities which are high enough to make neutron capture faster than β decay even for neutron excess nuclei 15–30 units from the stability line. These conditions enable the production of neutron-rich nuclei close to the dripline via neutron capture and (γ, n) photodisintegration during the r-process. Once the neutron source ceases, the progenitor nuclei decay either via β − or α emission or by fission towards stability and form the stable isotopes of elements up to the heaviest species Th, U and Pu. Due to the relatively small neutron separation energies in nuclei with Nmag + 1, where Nmag = 50, 82, 126, 184, the r-process flow at magic neutron numbers comes to a halt requiring several β decays to proceed. As the half lives of these magic nuclei are large compared to “regular” r-process nuclides, they determine the dynamical timescale of the r-process. Furthermore, much matter is accumulated at these ‘waiting points’ resulting in the observed peak structure in the r-process abundance distribution. As far from stability the masses A with magic neutron numbers are smaller, these abundance peaks are shifted relative to the s-process peaks. The r-process occurs under conditions for which an equilibrium between neutron captures and photodissociations is achieved as long as neutrons are available. 64 In this case, the path in the mass table followed by the r-process is mainly determined by the neutron separation energies. The r-process needs a precise knowledge of masses for thousand of nuclei reaching the neutron drip line. Traditionally nuclear masses are described empirically, but more recently microscopically based mass formulae have become available. Empirical mass formulae based on the nuclear liquid drop model have been improved by introducing phenomenologically microscopic corrections if needed to describe experimental data. Thus the macroscopic-microscopic mass formula, whose latest and most sophisticated version, the Finite Range Droplet Model (FRDM), has become the tool of choice and has been applied to many astrophysical problems, including r-process nucleosynthesis. As a number of merit, the FRDM mass formula reproduces the known masses of about 2000 nuclei with an rms deviation of about 0.7 MeV.65 The advances of (non-relativistic and relativistic) mean-field models and the progress in computational power have made it possible to develop microscopic mass models and apply them globally to the entire nuclear chart. The break-through of such microscopically founded model came with the development of the Extended Thomas–Fermi plus Strutinsky Integral (ETFSI) method.66 It reproduces the known nuclear masses as well as the FRDM approach, and has become the other standard model to produce the unknown nuclear masses in r-process simulations. Further progress in the development of microscopic mass formulae are mass compilations based on the shell model67 and more recently on HFB calculations with Skyrme forces fitted globally to nuclear masses. It was shown that the known nuclear masses can be reproduced with the same overall-quality as by the best

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FRDM fits. The microscopic approach has been extended to a full Hartree–FockBogoliubov treatment of the masses, assuming a specifically designed Skyrme force and a δ pairing force.68 Critical recent reviews of the status of nuclear masses both experimentally and theoretically can be found in.69,70 In addition to nuclear masses simulations of the r-process require the knowledge of beta-decay half-lives as they determine the flow of matter from the light seed nuclei to the heavier nuclei.71 The calculation of β decay half-lives usually requires two ingredients: the Gamow–Teller strength distribution in the daughter nucleus and the relative energy scale between parent and daughter (i.e. the Qβ value). Due to the huge number of nuclei relevant for the r process, the estimates of the half-lives are so far based on a combination of global mass models and the quasi particle randomphase approximation (see Ref. 27 for a description of the different models). However, recently shell-model calculations have become available for some key nuclei with a magic neutron number N = 50,27 N = 82,72,73 and N = 126.74 All this calculations suffer from the lack of spectroscopic information on the regions of interest that is necessary to fine tune the effective interactions. This situation is improving for N = 82 thanks to the recent spectroscopic data on 130 Cd.75 Moreover, for N = 126 β half-lives for nuclei approaching the r-process path are finally becoming available 76

10 −1

FRDM ETFSI−Q

10 −2 10 −3 10 −4

Y

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10 −5 10 −6 10 −7 10 −8 10 −9 60

90

120

150

180

210

240

A Fig. 3. Final r-process abundances (at 1.6 Gy) obtained in an adiabatic expansion model using two different mass models (FRDM65 and ETFSI-Q77 ). The solid circles correspond to a scaled solar r-process abundance distribution.78

Finally, if enough neutrons are present the r-process can reach regions where fission can take place. In this case fission cycling can contribute to obtain a robust r-process pattern as demanded by recent observations of metal-poor stars. 79 Figure 3

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shows the final r-process abundances resulting from two r-process calculations using identical astrophysical conditions but different nuclear input (masses and fission barriers) showing how sensitive are the resulting r-process abundances to different nuclear inputs. Acknowledgments It is a pleasure to thank our collaborators W.R. Hix, H.-Th. Janka, A. Juodagalvis, A. Kelić, K. Langanke, A. Marek, B. M¨ uller, I. Panov, J. M. Sampaio, K.-H. Schmidt, F.-K. Thielemann, N. T. Zinner and in particular T. Neff for providing much of the material for section 2. References 1. E. M. Burbidge, G. R. Burbidge, W. A. Fowler and F. Hoyle, Rev. Mod. Phys. 29, 547 (1957). 2. A. G. W. Cameron, Stellar Evolution, Nuclear Astrophysics, and Nucleogenesis, Report CRL-41, Chalk River (1957). 3. R. B. Wiringa and S. C. Pieper, Phys. Rev. Lett. 89, 182501 (2002). 4. S. C. Pieper and R. B. Wiringa, Ann. Rev. Nucl. Part. Sci. 51, 53 (2001). 5. P. Navr´ atil, J. P. Vary and B. R. Barrett, Phys. Rev. Lett. 84, 5728 (2000). 6. P. Navr´ atil, J. P. Vary and B. R. Barrett, Phys. Rev. C 62, 054311 (2000). 7. E. Epelbaum, Prog. Part. Nucl. Phys. 57, 654 (2006). 8. K. M. Nollet, R. B. Wiringa and R. Schiavilla, Phys. Rev. C 63, 024003 (2001). 9. K. M. Nollet, Phys. Rev. C 63, 054002 (2001). 10. P. Navr´ atil, C. A. Bertulani and E. Caurier, Phys. Lett. B 634, 191 (2006). 11. P. Navr´ atil, C. A. Bertulani and E. Caurier, Phys. Rev. C 73, 065801 (2006). 12. K. Langanke and H. Friedrich, in Adv. Nuc. Phys., eds. M. Baranger and E. Vogt (Plenum Press, New York, 1987) p. 223. 13. Y. Kanada-En’yo and H. Horiuchi, Prog. Theor. Phys. Suppl. 142, 205 (2001). 14. H. Feldmeier and J. Schnack, Rev. Mod. Phys. 72, 655 (2000). 15. R. Roth et al., Phys. Rev. C 72, 034002 (2005). 16. T. Neff and H. Feldmeier, Nucl. Phys. A 738, 357 (2004). 17. R. Roth, T. Neff, H. Hergert and H. Feldmeier, Nucl. Phys. A 745, 3 (2004). 18. A. Cribeiro, PhD thesis, TU Darmstadt2005. 19. T. Neff, H. Feldmeier and K. Langanke, Phys. Rev. C , p. (2007), “submitted”. 20. H. Schatz, L. Bildsten and A. Cummings, Astrophys. J. 583, L87 (2003). 21. L. R. Gasques et al., Phys. Rev. C 72, 025806 (2005). 22. T. Rauscher and F.-K. Thielemann, At. Data Nucl. Data Tables 75, 1 (2000). 23. K. Bennaceur et al., Phys. Lett. B 488, 75 (2000). 24. N. Michel et al., Phys. Rev. Lett. 89, 042502 (2002). 25. H. Herndl, J. G¨ orres, M. Wiescher, B. A. Brown and L. van Wormer, Phys. Rev. C 52, 1078 (1995). 26. J. L. Fisker, V. Barnard, J. G¨ orres, K. Langanke, G. Mart´ınez-Pinedo and M. C. Wiescher, At. Data. Nucl. Data Tables 79, 241 (2001). 27. K. Langanke and G. Mart´ınez-Pinedo, Rev. Mod. Phys. 75, 819 (2003). 28. G. M. Fuller, W. A. Fowler and M. J. Newman, Astrophys. J. Suppl. 42, p. 447 (1980). 29. G. M. Fuller, W. A. Fowler and M. J. Newman, Astrophys. J. 252, p. 715 (1982). 30. G. M. Fuller, W. A. Fowler and M. J. Newman, Astrophys. J. Suppl. 48, p. 279 (1982).

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58. A. Heger, E. Kolbe, W. Haxton, K. Langanke, G. Mart´ınez-Pinedo and S. E. Woosley, Phys. Lett. B 606, 258 (2005). 59. A. Juodagalvis, K. Langanke, G. Mart´ınez-Pinedo, W. R. Hix, D. J. Dean and J. M. Sampaio, Nucl. Phys. A 747, 87 (2005). 60. K. Langanke, G. Mart´ınez-Pinedo, P. von Neumann-Cosel and A. Richter, Phys. Rev. Lett. 93, 202501 (2004). 61. H.-T. Janka, K. Langanke, A. Marek, G. Martínez-Pinedo and B. M¨ uller, Phys. Repts. 442, 38 (2007). 62. K. Langanke, G. Martínez-Pinedo, B. M¨ uller, H.-T. Janka, A. Marek, W. R. Hix, A. Juodagalvis and J. Sampaio, Phys. Rev. Lett. , p. (2007), “submitted”. 63. J. J. Cowan and F.-K. Thielemann, Physics Today , 47(October 2004). 64. J. J. Cowan, F.-K. Thielemann and J. W. Truran, Phys. Repts. 208, 267 (1991). 65. P. M¨ oller, J. R. Nix and K.-L. Kratz, At. Data. Nucl. Data Tables 66, 131 (1997). 66. Y. Aboussir, J. M. Pearson, A. K. Dutta and F. Tondeur, At. Data Nucl. Data Tables 61, 127 (1995). 67. J. Duflo and A. P. Zuker, Phys. Rev. C 52, R23 (1995). 68. M. Samyn, S. Goriely and J. M. Pearson, Nucl. Phys. A 725, 69 (2003). 69. J. M. Pearson and S. Goriely, Nucl. Phys. A 777, 623 (2006). 70. D. Lunney, J. M. Pearson and C. Thibault, Rev. Mod. Phys. 75, 1021(Aug 2003). 71. B. Pfeiffer, K.-L. Kratz, F.-K. Thielemann and W. B. Walters, Nucl. Phys. A 693, 282 (2001). 72. B. A. Brown, R. Clement, H. Schatz, J. Giansiracusa, W. A. Richter, M. HjorthJensen, K. L. Kratz, B. Pfeiffer and W. B. Walters, Nucl. Phys. A 719, 177c (2003). 73. G. Mart´ınez-Pinedo and K. Langanke, Phys. Rev. Lett. 83, 4502 (1999). 74. G. Mart´ınez-Pinedo, Nucl. Phys. A 688, 357c (2001). 75. I. Dillmann, K.-L. Kratz, A. W¨ ohr, O. Arndt, B. A. Brown, P. Hoff, M. Hjorth-Jensen, U. K¨ oster, A. N. Ostrowski, B. Pfeiffer, D. Seweryniak, J. Shergur, W. B. Walters and “the ISOLDE Collaboration”, Phys. Rev. Lett. 91, 162503 (2003). 76. T. Kurtukian-Nieto et al., Phys. Rev. Lett. , p. (2007), “submitted”. 77. J. M. Pearson, R. C. Nayak and S. Goriely, Phys. Lett. B 387, 455 (1996). 78. J. J. Cowan, B. Pfeiffer, K.-L. Kratz, F.-K. Thielemann, C. Sneden, S. Burles, D. Tytler and T. C. Beers, Astrophys. J. 521, 194(August 1999). 79. G. Mart´ınez-Pinedo, D. Mocelj, N. Zinner, A. Kelić, K. Langankea, I. Panov, B. Pfeiffer, T. Rauscher and K.-H. S. F.-K. Thielemann, Prog. Part. Nucl. Phys. , p. (2007), in press.

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COUPLED-CLUSTER APPROACH TO AN AB-INITIO DESCRIPTION OF NUCLEI D. J. DEAN1,2 , G. HAGEN1,2,3 , M. HJORTH-JENSEN2,4 , and T. PAPENBROCK1,3 1 Physics

Division, Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, TN 37831, U.S.A.

2 Center

of Mathematics for Applications, University of Oslo, N-0316 Oslo, Norway 3 Department

of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, U.S.A.

4 Department

of Physics, University of Oslo, N-0316 Oslo, Norway

We presents results from ab-initio coupled-cluster theory for stable, resonant, and weakly bound nuclei. Results for the chain of helium isotopes 4−10 He, 16 O, and 40 Ca are discussed. Keywords: Nuclear structure calculations; Coupled-cluster theory; Weakly bound nuclei; Ground state resonances.

1. Introduction The theoretical description of bound, weakly bound, and unbound quantum manybody systems, together with present and planned experimental studies of such systems, represents a great challenge to our understanding of nuclear systems. Experiments in nuclear physics will address such important topics as how shells evolve, the role of many-body correlations, and the position of the stability lines of nuclei. The proximity of the scattering continuum in these systems implies that they should be treated as open quantum systems where coupling with the scattering continuum can take place. This means that a many-body formalism should contain resonant and continuum states in the basis in order to describe loosely bound systems or unbound systems. Extending the single-particle basis to include such degrees of freedom results in intractable dimensionalities for traditional configuration interaction methods (shell-model in nuclear physics) approaches. Shell-model codes tailored to the nuclear many-body problem can today reach dimensionalities of approximately 1010 basis states. Some of the systems studied here exhibit dimensionalities of some 1060 basis states. To circumvent this dimensionality problem, we have built a nuclear many-body program based on the coupled-cluster methods. Coupled-cluster theories allow for numerical cost-efficient ways of deal-

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ing with large dimensionalities compared with traditional configuration interaction methods. We report here new results from coupled-cluster theories including both bound, resonant, and continuum states.1–4 We also show that coupled-cluster theories reproduce benchmark results for light nuclei with minimal numerical cost and provide benchmarks for heavier nuclei.

2. Results and Discussion In addition to the above dimensionality problems, the nuclear many-body problem is riddled by the fact that there is no analytic expression for the underlying nucleonnucleon (NN) interaction. Furthermore, three-body interactions are important in nuclear physics and need to be included in a systematic way in a many-body formalism. In recent years, quite a lot of progress has been made within chiral effective field theories to construct NN and three-nucleon interactions from the underlying symmetries of QCD. The starting point is then a chiral effective Lagrangian with nucleons and pions as effective degrees of freedom only. Three-body interactions emerge naturally and have explicit expressions at every order in the chiral perturbation theory expansion. In this work we have chosen to work with a nucleonnucleon interaction derived from effective-field theory, such as the N3 LO model of Entem and Machleidt. In addition, we have also used the more phenomenological V18 interaction. We renormalize the short-range part of the nucleon-nucleon interaction by a similarity transformation technique in momentum space.4 This renormalized interaction defines our Hamiltonian which enters the solution of the coupled-cluster equations. To obtain ground-state energies of both bound and weakly bound systems, we need a many-body scheme which is (i) fully microscopic and size extensive, (ii) allows for inclusion, in a systematic way, various many-body correlations to be summed to infinite order, (iii) can account for the description of both closed-shell systems and valence systems, and (iv) capable to describe both bound and weakly bound systems. Coupled-cluster theories allow for the inclusion of all these features. Our coupled-cluster approaches include 1p − 1h and 2p − 2h correlations, normally dubbed single and double excitations (CCSD). Correlations of the 3p − 3h type are included perturbatively (labelled CCSD(T)) or via other approximations to the full 3p − 3h correlations (CCSDT). Furthermore, for weakly bound systems, we employ complex Gamow–Hartree–Fock single-particle basis and an effective interaction defined by such a single-particle basis.1,2 In the left panel of Fig. 1 we show the coupled-cluster results for 4 He and compare them with results from few-body calculations. There is excellent agreement, showing that coupled-cluster results reproduce other ab-initio results with a much smaller numerical cost. The right panel shows the corresponding results for 16 O, providing a benchmark for this nucleus. The results are given as a function of the number of

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oscillator shells, limited by 2n + l.2 See Ref. 2 for further details. The 16 O results show an overbinding, which most likely is due to omitted three-body interactions. Fig. 2 shows results for 16 O (left panel) and 40 Ca as a functions of the oscillator energy ~ω used in computing the oscillator wave function and the number of major shells N used in the coupled-cluster calculations. As expected, with increasing size of the model space, the results stabilize as a function of the chosen oscillator energy. Our results are converged with a given two-body Hamiltonian and we can therefore claim that lack of agreement with experiment is due to missing physics, such as three-body interactions, in our Hamiltonian. In Fig. 3 we present our recent CCSD results3 for the chain of helium isotopes using a complex single-particle basis. The largest model space has 850 single-particle orbitals, distributed among 5s5p5d4f 44h4i proton orbitals and 20s20p5d4f 44h4i neutron orbitals. For 10 He this results in approximately 1022 basic states. These are the first ever ab-initio calculations of weakly bound isotopes and we see that with a two-body Hamiltonian we are able to reproduce correctly the experimental trend and predict correctly which nuclei have bound ground states and which are resonances. The results are converged within our chosen model spaces. The quan-

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Fig. 1. Left figure is the binding energy for 4 He as a function of the number of oscillator shells N = 2n + l. The maximum orbital momentum was set to l = 7. The right panel exhibits the corresponding result for 16 O. Taken from Ref. 2.

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Fig. 2. Left figure is the binding energy for 16 O as function of the number of oscillator shells N = 2n + l and oscillator energy ~ω. The maximum orbital momentum was set to l = 7. The right panel is the corresponing result for 40 Ca. Taken from Ref. 2.

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Fig. 3. CCSD calculation of the ground states with the low-momentum N3 LO nucleonnucleon interaction for an increasing number of partial waves. Our calculated width of 10 He is ≈ 0.002MeV. TNF stands for three-body forces while triples are three-body correlations not included here. Taken from Ref. 3.

titative lack of agreement with experiment is due to our omission of three-body interactions. In summary, coupled-cluster theories hold great promise for a quantitative understanding of nuclei. With the possibility to include three-body interactions,1 we may be able to tell how nuclei evolve as one moves towards the drip line. Acknowledgments This work was supported in part by the U.S. Department of Energy under Contract Nos. DE-AC05-00OR22725 (Oak Ridge National Laboratory), DE-FG0296ER40963 (University of Tennessee), DE-FG05-87ER40361 (Joint Institute for Heavy Ion Research), DE-FC02-07ER41457 (University of Washington) and by the Research Council of Norway (Supercomputing grant NN2977K). Computational resources were provided by the Oak Ridge Leadership Class Computing Facility and the National Energy Research Scientific Computing Facility. Discussions with A. Schwenk are acknowledged. References 1. G. Hagen, T. Papenbrock, D. J. Dean, A. Schwenk, M. Wloch, P. Piecuch, and A. Nogga, Phys. Rev. C 76, 034302 (2007). 2. G. Hagen, D. J. Dean, M. Hjorth-Jensen, T. Papenbrock, and A. Schwenk, Phys. Rev. C 76, in press (2007). 3. G. Hagen, D. J. Dean, M. Hjorth-Jensen, and T. Papenbrock,Phys. Lett. B 655, in press (2007). 4. G. Hagen, M. Hjorth-Jensen, and N. Michel, Phys. Rev. C 73, 064307 (2006).

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DEVELOPING NEW MANY-BODY APPROACHES FOR NO-CORE SHELL MODEL CALCULATIONS B. R. BARRETT∗ and A. F. LISETSKIY Department of Physics, University of Arizona, Tucson, AZ 85721, USA ∗ E-mail: [email protected] ` P. NAVRATIL Lawrence Livermore National Laboratory, Livermore, CA 94550, USA E-mail: [email protected] I. STETCU Los Alamos National Laboratory, Los Alamos, NM 87845, USA E-mail: [email protected] J. P. VARY Department of Physics and Astronomy, Iowa State University, Ames, IA 50011 E-mail: [email protected] We present a method to derive sd-shell effective interactions from the No-Core Shell Model effective interaction by taking into account many-body correlations. The properties of the derived effective interaction are analyzed. Keywords: Shell model; effective interactions; nuclear structure.

1. Introduction The No-Core Shell Model (NCSM) has had considerable success in describing the binding energies, excitation spectra and other physical properties of light nuclei, A≤16 e.g.1 One of the principal reasons for these successes is that one has a welldefined procedure for calculating the effective interactions and operators to be used in a given model space, using a unitary transformation approach in a given cluster approximation, e.g., usually for two-body or three-body clusters. Recent NCSM investigations have included the three-body components of the effective interaction, 2 i.e., the three-body cluster; have studied the effects associated with three-nucleon (NNN) interactions;3,4 have utilized chiral (NN) and (NNN) nuclear interactions;5,6 have implemented the Lorentz integral transform method;7 and have linked nuclear structure with reactions8 in applications to p-shell nuclei. The big challenge facing future NCSM investigations is now to perform such calculations for heavier nuclei, for which the model spaces become unmanageable

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for existing computer power. Our current studies involve the development of new many-body approaches for achieving this goal. In this contribution we present a method to derive sd-shell effective interactions from the NCSM effective interaction by taking into account many-body correlations. The properties of the derived effective interaction are analyzed. 2. NCSM and sd-shell Effective Interactions The starting point of the NCSM approach is the bare, relative (i.e., translationally invariant) A-body Hamiltonian constrained by the Center-of-Mass Harmonic Oscillator (HO) potential.1 The eigenvalue problem for the exact A-body Hamiltonian is complicated technically for A ≥ 3 since a huge configurational space is required (Nmax → ∞), where Nmax is the maximal number of excited HO quanta. However the problem considerably simplifies in the (a=2)-body cluster approximation.1 Using an HO basis, one finds that the value of Nmax = 450 is a very good approximation for the Nmax → ∞, in solving the two-nucleon problem, and is sufficient to take account of the strong short-range correlations. One can then employ this Nmax ,Ω,eff 2-body cluster approximation to construct the effective HA,a=2 Hamiltonian for small values of Nmax via a unitary transformation U2 , Nmax ,Ω,eff Ω HA,2 = U2 HA,2 U2† ,

(1)

Ω Hamiltonian in the full so that some subset of the eigenvalues of the bare HA,2 Nmax ,Ω,eff A-body space are exactly reproduced by HA,a=2 in the smaller model space defined by Nmax . Following the prescription outlined above, we have derived effective two-body interactions for the A = 18 system in 2~Ω (Nmax = 2) and 4~Ω (Nmax = 4) spaces using Argonne V18 (AV18) NN potential.9,10 Nmax ,Ω The corresponding 18-body Hamiltonian HA=18,a=18 can be represented in terms

Nmax ,Ω,eff of the 2-body effective HA=18,a=2 Hamiltonian and diagonalized. Using the idea proposed in Navr´ atil et al.,14 one can use the eigenvalues and Nmax ,Ω eigenvectors of the Hamiltonian HA=18,a=18 to derive the effective A-body Hamiltonian for the Nmax = 0 model space, which reproduces exactly the lowest dN =0 Nmax ,Ω eigenvalues of the Hamiltonian HA=18,a=18 . To construct this effective Hamiltonian the same procedure given by the Eq. (1) can be used, where the two-body unitary transformation U2 is to be replaced with the 18-body unitary transformation U18 Nmax ,Ω constructed using 18-body eigenvectors of the Hamiltonian HA=18,a=18 . In the considered Nmax = 0 case the dimension of the 18-body effective Hamiltonian is the same as the dimension of the 2-body Hamiltonian in the sd-space. This means that the effective Hamiltonian contains only 1-body and 2-body terms even after the exact 18-body cluster U18 transformation. The s- and p-spaces are fully occupied by the other 16 nucleon-spectators and the total 18-body wave function can be exactly factorized as the wave function of the 0+ 16 O core times the 2-body sd-space wave function. This allows one to determine the effective two-body Hamiltonian for the

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Fig. 1. The excitation energies of the JTπ states for 18 F calculated in the 4~Ω space with the AV189,10 (for ~Ω = 15 MeV), N3 LO11 (for ~Ω = 14 MeV) and CD-Bonn12 (for ~Ω = 15 MeV) potentials. The USD13 spectra is shown for comparison.

sd-subspace. We have calculated effective sd-space Hamiltonians for 18 F, using the many-body 2~Ω and 4~Ω Hamiltonians constructed from the AV18 interaction.9,10 Nmax =0,Ω,eff The resulting HA=18,a=18 Hamiltonian reproduces exactly the excitation energies of the lowest 28 sd-space dominated states of 4~Ω NCSM calculations for 18 F, as shown in Fig. 1. To calculate the spectra of other A > 18 “sd-shell” nuclei, we have decomposed Nmax =0,Ω,eff the effective HA=18,a=18 Hamiltonian into one-body and pure two-body parts. The effective one-body part, single particle d5/2 , d3/2 and s1/2 energies, are taken as the + + 17 corresponding excitation energies of the 5/2+ F calculated 1 , 3/21 , 1/21 states of in the 2~Ω and 4~Ω spaces for corresponding interactions. As an example, the calculated spectra of 20 Ne within the Standard Shell Model (SSM) using the derived effective interaction and within the NCSM using the corresponding 2hΩ interaction are compared in Fig. 2. We notice that the SSM almost perfectly matches the 2~Ω NCSM excitation energies, although the dimension of the 2~Ω space for 20 Ne approaches 542 072 (m-scheme) and the dimension of the sd-space is only 640. Apparently, the observed small differences for the excitation energies may be attributed to the three- and four-body sd-space correlations as well as the density dependence, which are not taken into account in the derived effective two-body interaction. The comparison of the derived sd-shell effective and corresponding original interactions for different values of Nmax will help also to establish a direct procedure for constructing effective interactions that takes into account many-body correlations.

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0+

Fig. 2. SSM and NCSM for 20 Ne with sd-shell effective (2AV18SD) and original 2~Ω AV18 (2AV18) interactions, respectively. Experimental and USD spectra are shown for comparison.

Acknowledgments This research supported in part by NSF grant No. PHY0555396 (B.R.B., A.F.L.), by DOE grant No.DE-AC52-06NA25396 (I.S.), and by DOE grant No. DE-FG0287ER40371 (J.P.V.). This work in part performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344 (P.N.). References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

P. Navratil, J.P.Vary, B.R.Barrett, Phys. Rev. C 62, 054311 (2000). P. Navratil and W.E.Ormand, Phys. Rev. Lett. 88, 152502 (2002). P. Navratil and W.E.Ormand, Phys. Rev. C 68, 034305 (2003). A. C. Hayes, P.Navratil, J.P.Vary et al., Phys. Rev. Lett. 91, 012502 (2003). A. Nogga, P.Navratil, B.R.Barrett, J.P.Vary, Phys. Rev. C 73, 064002 (2006). P. Navr´ atil, V. G. Gueorguiev, J. P. Vary, W. E. Ormand, and A. Nogga, Phys. Rev. Lett. 99, 042501 (2007). I. Stetcu, S.Quaglioni, S.Bacca, B.R.Barrett, C.W.Johnson, P.Navratil, N.Barnea, W.Leidemann, G.Orlandini, Nucl. Phys. A785, 307 (2007). P. Navratil, C.A.Bertulani, E.Caurier, Phys. Lett. B634, 191 (2006). R. B. Wiringa, V. G. J. Stocks, and R. Schiavilla, Phys. Rev. C 51, 38 (1995). S. Pieper and R. B. Wiringa, Annu. Rev. Nucl. Part. Sci. 51, 53 (2001). D. R. Entem and R. Machleidt, Phys. Rev. C 68, 041001(R) (2003). R. Machleidt, F. Sammarruca, and Y. Song, Phys. Rev. C 53, 1483 (1996). B. A. Brown and B. H. Wildenthal, Ann. Rev. of Nucl. Part. Sci. 38, 29 (1988). P. Navratil, M. Thoresen, and B. R. Barrett, Phys. Rev. C 55, R573 (1997).

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APPLICATIONS OF IN-MEDIUM CHIRAL DYNAMICS TO NUCLEAR STRUCTURE P. FINELLI Department of Physics, University of Bologna, Bologna, 40126, Italy E-mail: [email protected] http : //www − th.bo.inf n.it/activities/N uclear P hysics/f inelli.html A relativistic nuclear energy density functional is developed, guided by two important features that establish connections with chiral dynamics and the symmetry breaking pattern of low-energy QCD: a) strong scalar and vector fields related to in-medium changes of QCD vacuum condensates; b) the long- and intermediate-range interactions generated by one-and two-pion exchange, derived from in-medium chiral perturbation theory, with explicit inclusion of ∆(1232) excitations. The results are at the same level of quantitative comparison with data as the best phenomenological relativistic mean-field models. Keywords: Chiral dynamics; density functional theory.

1. Introduction One of the most complete and accurate description of structure phenomena in finite nuclei is currently provided by self-consistent non-relativistic and relativistic meanfield approaches. They represent an approximate implementation of Kohn–Sham density functional theory (DFT).1 The DFT provides a description of the nuclear many-body problem in terms of an energy density functional, E[ρ]. A major goal of nuclear structure theory is to build an energy density functional which is universal, in the sense that the same functional is used for all nuclei, with the same set of parameters. This framework should then provide a reliable microscopic description of infinite nuclear and neutron matter, ground-state properties of bound nuclei, rotational spectra and low-energy vibrations. In order to formulate a microscopic nuclear energy density functional, one must be able to go beyond the mean-field approximation and systematically calculate the exchange-correlation part, E xc [ρ], of the energy functional, starting from the relevant active degrees of freedom at low energy. The exact Exc includes all many-body effects; the usefulness of DFT crucially depends on our ability to construct accurate approximations to the exact exchange-correlation energy. The natural microscopic framework is chiral effective field theory.2 Our approach to the nuclear energy density functional, emphasizing links with

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low-energy QCD and its symmetry breaking pattern, has been introduced in Refs. 3 and 4 and it is based on the following conjectures: 1) The nuclear ground state is characterized by strong scalar (US ) and vector (UV ) mean fields which have their origin in the in-medium changes of the scalar quark condensate (the chiral condensate) and of the quark density. They can be calculated by QCD sum rules techniques5 to obtain, at leading order, σN M N ρS , m2π fπ2 4(mu + md )MN = ρ, m2π fπ2

US = −

(1)

UV

(2)

where σN = hN |mq q¯q|N i is the nucleon sigma term (' 50 MeV), mπ is the pion mass (138 MeV), fπ = 92.4 MeV is the pion decay constant and ρ and ρS are the baryon and the scalar density, respectively. The resulting US and UV are individually of the order of 300 − 400 MeV in magnitude. Their ratio US σN ρS =− UV 4(mu + md ) ρ

(3)

is close to −1. As a result, in the single-nucleon Dirac equation there is an almost complete cancellation in the central potential (∼ UV + US ), giving a negligible contribution to the binding of the system, but, at the same time, a large contribution to the spin-orbit potential 1 1 ∂ (UV − US ) l · s . VLS ∼ 2m2 r ∂r 2) Nuclear binding and saturation arise primarily from chiral (pionic) fluctuations in combination with Pauli blocking effects and three-nucleon (3N) interactions, superimposed on the condensate background fields and calculated according to the rules of in-medium chiral perturbation theory (ChPT). The starting point is the description of nuclear matter based on the chiral effective Lagrangian with pions and nucleons with the inclusion of explicit ∆(1232) degrees of freedom.6 The relevant small scales are the Fermi momentum kf , the pion mass mπ and the ∆ − N mass difference ∆ ≡ M∆ − MN ' 2.1mπ , all of which are well separated from the characteristic scale of spontaneous chiral symmetry breaking, 4πfπ ' 1.16 GeV with the pion decay constant fπ = 92.4 MeV. The calculations have been performed to three-loop order in the energy density. They incorporate the one-pion exchange Fock term, iterated one-pion exchange and irreducible two-pion exchange, including one or two intermediate ∆’s. The expansion coefficients are functions of kf /mπ and ∆/mπ , the dimensionless ratios of the relevant small scales. Divergent momentum space loop integrals are regularized by introducing subtraction constants in the spectral representations of these terms (the only parameters in this approach). They encode short-distance dynamics not resolved in detail at the characteristic momentum scale kf 4πfπ . The finite parts of the energy density, written in closed form as

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200 p [MeV]

300

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Fig. 1. Energy per particle of symmetric nuclear matter (upper left) and the asymmetry energy (upper right) as functions of the nucleon density. Energy per particle of pure neutron matter (lower left) as a function of neutron density, and the momentum dependence of the real part of the single-nucleon potential at nuclear saturation (lower right). The dashed curves refer to the results obtained with only pions and nucleons as active degrees of freedom. The solid curves include the contribution from two-pion exchange with single and double virtual ∆(1232)-isobar excitations.

functions of kf /mπ and ∆/mπ , represent long and intermediate range (chiral) dynamics with input fixed entirely in the πN sector. The low-energy constants (contact terms) are adjusted to reproduce basic properties of symmetric and asymmetric nuclear matter (see Fig. 1).

2. Model The relativistic density functional describing the ground-state energy of the system can be written as a sum of four distinct terms: Z ¯ Efree [ˆ ρ] = d3 r hφ0 |ψ[−iγ · ∇ + MN ]ψ|φ0 i , (4) Z 1 (0) ¯ 2 (0) ¯ 2 d3 r {hφ0 |GS (ψψ) |φ0 i + hφ0 |GV (ψγ (5) EH [ˆ ρ] = µ ψ) |φ0 i} , 2 Z n 1 (π) ¯ 2 |φ0 i + hφ0 |G(π) (ˆ ¯ µ ψ)2 |φ0 i Eπ [ˆ ρ] = d3 r hφ0 |GS (ˆ ρ)(ψψ) ρ)(ψγ V 2 (π) ¯τ ψ)2 |φ0 i + hφ0 |G(π) (ˆ ¯ τ ψ)2 |φ0 i +hφ0 |GT S (ˆ ρ)(ψ~ T V ρ)(ψγµ ~ o (π) ¯ 2 |φ0 i , − hφ0 |DS [∇(ψψ)] (6) Z 1 1 + τ3 Ecoul [ˆ ρ] = d3 r hφ0 |Aµ eψ¯ γµ ψ|φ0 i , (7) 2 2

where |φ0 i denotes the nuclear ground state. Here Efree is the energy of the free (relativistic) nucleons including their rest mass. EH is a Hartree-type contribution representing strong scalar and vector mean fields, later to be connected with the

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A Fig. 2. The deviations (in percent) of the calculated binding energies from the experimental values of N d, Sm, Gd, Dy, Er, Y b, Hf , Os, and P t isotopes. We used the Gogny interaction in the pairing channel.

leading terms of the corresponding nucleon self-energies deduced from in-medium QCD sum rules. Furthermore, Eπ is the part of the energy generated by chiral πN ∆dynamics, including a derivative (surface) term, with all pieces explicitly derived in Ref. 6. (0) The couplings are decomposed into density-independent parts Gi which arise from strong isoscalar scalar and vector background fields, and density-dependent (π) parts Gi (ˆ ρ) generated by (regularized) one- and two-pion exchange dynamics. It (π) is assumed that only pionic processes contribute to the isovector channels. DS is a surface term and can be derived within the chiral approach.6 To demonstrate that chiral effective field theory provides a consistent microscopic framework for finite nuclei description, we show in Fig. 2 a large set of calculations for isotope chains of deformed nuclei. Good agreement is found over the entire region of deformed nuclei. The maximum deviation of the calculated binding energies from data is below 0.5% for all isotopes. Acknowledgment I would like to thank my collaborators Dario Vretenar, Norbert Kaiser and Wolfram Weise. This work has been supported by MIUR and INFN. References 1. 2. 3. 4.

R. M. Dreizler and E. K. U. Gross, Density Functional theory, Spinger-Verlag, 1990. S. Scherer and M. R. Schindler, arXiv:hep-ph/0505265. P. Finelli, N. Kaiser, D. Vretenar and W. Weise, Nucl. Phys. A 735 p. 449 (2004). P. Finelli, N. Kaiser, D. Vretenar and W. Weise, Nucl. Phys. A 770 p. 1 (2006).

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5. T. D. Cohen, R. J. Furnstahl, D. K. Griegel and X. M. Jin, Prog. Part. Nucl. Phys. 35 p. 221 (1995). 6. S. Fritsch, N. Kaiser and W. Weise, Nucl. Phys. A 750 p. 259 (2005).

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VARIATIONAL CALCULATIONS OF THE EQUATION OF STATE OF NUCLEAR MATTER M. TAKANO∗ and H. KANZAWA Waseda University, 3-4-1 Okubo Shinjuku-ku, Tokyo 169-8555, Japan ∗ E-mail: [email protected] K. OYAMATSU Aichi Shukutoku University, Nagakute-cho, Aichi 480-1197, Japan K. SUMIYOSHI Numazu Sollege of Technology, Ooka 3600, Numazu, Shizuoka 410-8501, Japan We construct the equation of state (EOS) for infinite nuclear matter at zero and finite temperatures with the variational method starting from the realistic nuclear Hamiltonian composed of the Argonne V18 two-body potential and the UIX three-body interaction (TNI). At zero temperature, we evaluate the expectation value of the two-body nuclear Hamiltonian using the Jastrow-type wave function in the two-body cluster approximation with two conditions: The extended Mayer’s condition and the healing-distance condition. Then we take into account the TNI contribution which includes adjustable parameters whose values are determined so as to reproduces the empirical saturation data. The maximum mass of the neutron star with the present nuclear EOS is 2.2 M . At finite temperatures, we employ a method by Schmidt and Pandharipande, to obtain the free energy for nuclear matter. The critical temperature is about 18 MeV. We also calculate the free energy for asymmetric nuclear matter. Keywords: Nuclear matter; Nuclear EOS; variational method; neutron stars; supernovae.

1. Introduction The nuclear equation of state (EOS) plays important roles for astrophysical study such as supernovae (SN), hypernovae and neutron-star mergers. At present, however, there are only few nuclear EOSs available for SN simulations. Since these EOSs are constructed using phenomenological models, a SN-EOS based on the microscopic many-body approach is desirable. Considering these situations, we undertake to construct a new nuclear EOS for SN simulations using the variational calculations staring from the realistic nuclear forces. In this paper, as the first step of this study, we calculate the energies for uniform nuclear matter at zero and finite temperatures.1 In the next section, we calculate the energy at zero temperature, and then extend the calculation to finite temperatures in Sec. 3.

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2. Variational Calculations at Zero Temperature We start from the nuclear Hamiltonian composed of the AV18 two-body nuclear potential and the UIX three-body nuclear potential. We separate the Hamiltonian into H2 and H3 ; the former is the part without the three-nucleon interaction (TNI) and the latter is the TNI part. We assume the Jastrow-type trial wave function;   Y Ψ = Sym  fij  ΦF , (1) i>j

with the correlation function fij being expressed as fij =

1 X 1 X t=0 s=0

[fCts (rij ) + sfTt (rij )STij + sfSOt (rij )s · Lij ] Ptsij .

(2)

where fCts (r), fTt (r) and fSOt (r) are the spin-isospin-dependent central, tensor and spin-orbit correlation functions, respectively. In Eq. (2), ST is the tensor operator, s · Lij is the spin-orbit operator and Ptsij is the spin-isospin projection operators. With use of this wave function, we calculate the expectation value of H2 in the two-body cluster approximation, and denote it as E2 . Then, we minimize E2 /N with respect to fCts (r), fTt (r) and fSOt (r) by solving the Euler–Lagrange equations with the following two constraints. One is the extended Mayer’s conditions. The other is the healing distance condition: we impose that, at r ≥ rh , the correlation vanishes, i.e., fCts (r) = 1, fTt (r) = 0 and fSOt (r) = 0. Here, we assume that rh is proportional to r0 = (3/(4πρ))1/3 with the coefficient ah being chosen so that the calculated E2 /N for symmetric nuclear matter (SNM) is close to the result of the Fermi Hypernetted Chain (FHNC) calculations by Akmal, Pandharipande and Ravenhall (APR).2 The parameter ah obtained in this way is ah =1.76, and E2 /N for SNM and for pure neutron matter (PNM) are in good agreement with those by APR. We note that the healing distance condition plays an important role for reproduction of the results by APR. The contribution of H3 is taken into account as follows. First, we decompose H3 into two parts, the two-π-exchange part, H32π , and the repulsive part, H3R . Then, we evaluate the expectation values of them using the Fermi gas wave function, hH32π iF and hH3R iF , respectively. Using these terms, the TNI energy E3 /N is expressed as E3 /N = αhH3R iF /N + βhH32π iF /N + γρ2 exp[−δρ]

(3)

where α, β, γ and δ are adjustable parameters whose values are determined so that the total energy E/N = E2 /N + E3 /N reproduces the empirical saturation density ρ0 , saturation energy E0 /N , incompressibility K and symmetry energy Esym /N . We note that α and β are common to SNM and PNM, while γ is set to be zero for PNM. The obtained energies E/N for PNM and SNM are in fair agreement with those by APR as shown in Fig. 1, with ρ0 = 0.16fm−3 , E0 /N = −15.8 MeV, K = 250 MeV and Esym = 30 MeV. The maximum mass of the neutron star using the present result is about 2.2 M .

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3. Variational Calculations at Finite Temperatures For nuclear matter at finite temperatures, we employ the procedure by Schmidt and Pandharipande.3 In this method, the free energy per nucleon F/N is expressed as F/N = E0T /N − T S0 /N,

(4)

where E0T /N is the approximate internal energy and S0 /N is the approximate entropy at temperature T . E0T /N is the sum of E2T /N and E3 /N . E2T /N is the two-body cluster approximation of the expectation value of H2 using the Jastrowtype wave function at finite temperature, whose form is similar to that at zero temperature as shown in Eq. (1) with the single-particle wave function ΦF being specified by the averaged occupation probability n(k). For simplicity, E3 /N is chosen to be the same as at zero temperature. The approximate entropy S0 /N is expressed using n(k) as in the case of the Fermi gas. Here, the averaged occupation probability n(k) is expressed as −1 (k) − µ0 , (5) n(k) = 1 + exp kB T

with (k) = ~2 k 2 /(2m∗ ) being the quasi-particle energy including the effective mass of the nucleon m∗ . µ0 is determined by the normalization conditions. Then, the total free energy F/N is minimized with respect to m∗ . The obtained free energies are shown in Fig. 1, and the critical temperature is about 18 MeV. We note that the energies at zero and finite temperatures are treated in a consistent way. Furthermore, the internal energy and the entropy derived from F/N are very close to E0T /N and S0 /N , which implies that the present variational calculation is self-consistent. Furthermore, we calculate the free energies for asymmetric nuclear matter. At each proton fraction x, the correlation functions of three isospin-triplet states are

100

F/N[MeV]

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T = 0 MeV T = 10 MeV T = 20 MeV PNM (APR) SNM (APR)

0

-50 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ρ[fm−3] Fig. 1.

Free energies for neutron matter and symmetric nuclear matter.

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0 -10 -20 -30 0.0

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x Fig. 2. Free energies for asymmetric nuclear matter as functions of the proton fraction x at ρ = 0.16fm−3 .

treated independently for the energy calculations at zero temperature. The averaged occupation probabilities for protons and neutrons are also treated separately, and the free energy is minimized with respect to the effective masses for proton and for neutron. The results are shown in Fig. 1. The free energy increases quadratically as the proton fraction x decreases. We are planning to treat inhomogeneous nuclear matter using the Thomas–Fermi approximation to construct the nuclear EOS suitable for SN simulations. Acknowledgments This study is supported by a Grant-in-Aid for the 21st century COE program “Holistic Research and Education Center for Physics of Self-organizing Systems” at Waseda University, and Grant-in-Aid from the Scientific Research Fund of the JSPS (Nos. 18540291, 18540295 and 19540252). A part of the numerical calculations were performed with SR8000/MMP and SR11000/J2 at the Information Technology Center of the University of Tokyo. References 1. H. Kanzawa, K. Oyamatsu, K. Sumiyoshi and M. Takano, Nucl. Phys. A 791, 232 (2007). 2. A. Akmal, V. A. Pandharipande and D. G. Ravenhall, Phys. Rev. C 58, 1804 (1998). 3. K. E. Schmidt and V. R. Pandharipande, Phys. Lett. B 87, 11 (1979).

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REFINEMENT OF THE VARIATIONAL METHOD WITH APPROXIMATE ENERGY EXPRESSIONS BY TAKING INTO ACCOUNT 4-BODY CLUSTER TERMS K. TANAKA∗ and M. TAKANO Research Institute for Science and Engineering, Waseda University, 3-4-1 Okubo Shinjuku-ku, Tokyo 169-8555, JAPAN ∗ E-mail: [email protected] The approximate energy expression for neutron matter is refined by properly taking into account tensor correlations. The energy expression is an explicit functional of two-body distribution functions and used conveniently in the variational method. The previously proposed energy expression does not include the kinetic energy caused by the noncentral correlation sufficiently. In this study, important missing three-body-cluster kinetic-energy terms that are related to the tensor correlations are introduced into the energy expression. The refined energy expression automatically guarantees necessary conditions on tensor structure functions. Furthermore, the main part of the four-body-cluster kinetic-energy terms caused by the central and tensor correlations are included in the refined energy expression properly. The numerical results using the refined energy expression show significant improvement. Keywords: Neutron matter; variational method; tensor correlations.

1. Introduction We have been studying the variational method with approximate energy expressions for nuclear matter.1,2 The energy expression is an explicit functional of two-body distribution functions. Then, we can derive the Euler–Lagrange equations for these functions to obtain fully minimized energies. If we consider the central force only, the numerical results are reasonable.1 However, when the noncentral forces are taken into account, the obtained energies are too low and the noncentral distribution functions have unrealistically long tails, which implies that the kinetic energy caused by the noncentral correlations are not included in the energy expression appropriately. 2 Therefore, as the first step of the refinement, we improve the energy expression with respect to the noncentral correlations, especially the tensor correlation. In the next section, we introduce the energy expression, and refine it in Sec. 3. Numerical results and discussions are given in Sec. 4. 2. Approximate Energy Expression for Neutron Matter We consider spin-unpolarized neutron matter at zero-temperature and start from the following Hamiltonian,

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H =−

N N X ~2 2 X ∇i + Vij . 2m i<j i

(1)

The two-body potential Vij is chosen as the V14-type: ( 1 X Vij = VCs (rij ) + s [VT (rij )STij + VSO (rij )s · Lij ] + VqLs (rij )L2ij s=0

+ sVqSO (rij )(s · Lij )

2

)

Psij .

(2)

Here, Psij are spin-projection operators, STij is the tensor operator and Lij is the orbital angular momentum operator. In order to construct the energy expression, we introduce the central, tensor and spin-orbit distribution functions, as well as the structure functions, XZ Ψ † (x1 , · · ·, xN )Ps12 Ψ (x1 , · · ·, xN )dr 3 · · · dr N , (3) Fs (r12 ) = Ω 2 spin

FT (r12 ) = Ω 2

XZ

spin

FSO (r12 ) = Ω 2

XZ

spin

Ψ † (x1 , · · ·, xN )ST12 P112 Ψ (x1 , · · ·, xN )dr 3 · · · dr N ,

(4)

Ψ † (x1 , · · ·, xN )(s · L12 )P112 Ψ (x1 , · · ·, xN )dr 3 · · · dr N ,

(5)

2 + * N X exp(ik · r i ) = 1 + S1 (k) + S0 (k) ≥ 0, i=1 2 + * N X 1 1 σ i exp(ik · r i ) = 1 + S1 (k) − S0 (k) ≥ 0, SC2 (k) ≡ 3N i=1 3

1 SC1 (k) ≡ N

(6)

(7)

with Ss (k) being the Fourier transforms of Fs (r). Using these functions, the energy expression is constructed in Ref. 2 as # ) Z ∞ ("X 1 E 3 = EF + 2πρ Fs (r)VCs (r) + FT (r)VT (r) + FSO (r)VSO (r) r2 dr N 5 0 s=0 # ) Z ∞ ("X 1 2 X ECn 2 FqLs (r)VqLs (r) + FqSO (r)VqSO (r) r dr + + 2πρ N 0 s=0 n=1 ( ) 2 Z 1 π~2 ρ ∞ X 1 dFFs (r) 1 dFCs (r) + FCs (r) − r2 dr 2m 0 s=0 FCs (r) dr FFs (r) dr # ) 2 2 Z " ( 6 π~2 ρ ∞ 2 dgSO (r) dgT (r) 2 + 2 [gT (r)] FF1 (r)+ FqF1 (r) r2 dr, + 8 2m 0 dr r 3 dr (8)

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with ECn ~2 =− N 16π 2 mρ

Z

∞ 0

2

(2n − 1)

[SCn (k) − 1] [SCn (k) − SCF (k)] 4 k dk. SCn (k)/SCF (k)

(9)

In Eq. (8), FCs (r), gT (r) and gSO (r) are the auxiliary functions whose definitions are given in Ref. 2. We note that this energy expression automatically guarantees necessary conditions on SCn (k), Ineqs. (6) and (7). 3. Refinement of the Approximate Energy Expression We refine the energy expression by taking into account the tensor correlation. Since the main part of the potential energy expression is exact, we consider the refinement of the kinetic energy terms. First we assume the Jastrow wave function,   Y (10) Ψ (x1 , . . . , xN ) = Sym  fij  Φ(x1 , . . . , xN ). i<j

Here, Sym is the symmetrizer, Φ is the Fermi-gas wave function and fij is the correlation function expressed as fij =

1 X

[fCs (rij ) + sfT (rij )STij ] Psij ,

(11)

s=0

where fCs (r) and fT (r) are the central and tensor correlation functions, respectively. Using the Jastrow wave function, we cluster-expand both the expectation values of the Hamiltonian hHi/N and the energy expression E/N . Then, it is seen that the one body and two body cluster terms in hHi/N are completely included in E/N , while the three body and higher cluster terms are not. The main part of the missing three body cluster terms are expressed using the structure functions as Z ∞ Z ∞ ~2 ~2 E3CT 2 4 [S (k) − 1] [S (k)] k dk + [ST (k)]3 k 4 dk. =− C2 T N 32mπ 2 ρ 0 576mπ 2 ρ 0 (12) with ST (k) being defined as follows: Z ∞ ST (k) = 4πρ FT (r)j2 (kr)r2 dr. (13) 0

We note that E3CT /N goes to negative infinity through the variational procedure; it is harmful. Therefore, we convert this harmful term into harmless one using the tensor structure functions defined as 2 + * N X 1 1 SCT1 (k) ≡ (σ · k) exp(ik · r ) = SC2 (k) − ST (k) ≥ 0, (14) i i N k 2 i=1 3 2 + * N X 1 1 = SC2 (k) + ST (k) ≥ 0. (15) (σ × k) exp(ik · r ) SCT2 (k) ≡ i i 2N k 2 i=1 6

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New Old Central only

20 E/N[MeV]

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10

0 0.0

0.2

0.4 ρ[fm−3]

0.6

0.8

Fig. 1. Energy per neutron for neutron matter with the AV18 potential. The solid curve is for refined energy expression and the dashed curve is for old energy expression. The energy with the central force only is also shown by the dot-dashed curve.

Then, we propose the refined energy expression Enew /N as the right hand side of Eq. (8) with EC2 /N being replaced by 2 Z ∞ 2 X [SCTn (k) − 1] [SCTn (k) − SCF (k)] 4 ~2 EC2new n =− k dk. N 16π 2 mρ n=1 0 SCTn (k)/SCF (k)

(16)

Here, we introduced the denominators in the integrands in Eq. (16) not only for conversion of the harmful terms to the harmless ones but also for guarantee of the necessary conditions on SCTn (k). Furthermore, these denominators correspond to the four body and higher order cluster contributions. In fact, we confirmed that at least the main part of the four-body cluster terms in hHi/N , i.e., the direct terms that are the lowest order in the correlation, are included in Enew /N properly. 4. Numerical Results and Discussion Using the isoscalcar part of the AV18 potential as the two-body nuclear potential Vij , we perform the variational calculations for neutron matter. As shown in Fig. 1, Enew /N is higher than the old E/N significantly. In this calculation, the unrealistically long tail of FT (r) disappears. Also shown in Fig. 1 is the energy in which only the central component of the AV18 potential is taken into account. Enew /N is still much lower than the energy with the central force only; further improvement of the kinetic energy expression caused by the spin-orbit correlation is expected to reduce this energy difference. We also note that, in the calculation using the old energy expression, the obtained SCT2 (k) violate Ineq. (15), which implies that the necessary conditions on SCTn (k) play important roles in this variational calculation. We are now refining the energy expression for symmetric nuclear matter in a similar way. Refinement of the energy expression with respect to the spin-orbit correlation is our future problem.

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Acknowledgments This study is supported by a Waseda University Grant for Special Research Projects 2003A-628 and a Grant-in-Aid for the 21st century COE program “Holistic Research and Education Center for Physics of Self-organizing Systems” at Waseda University. References 1. M. Takano and M. Yamada, Prog. Theor. Phys. 91, 1149 (1994). 2. M. Takano and M. Yamada, Prog. Theor. Phys. 100, 745 (1998).

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COMPUTATIONAL QUANTUM MANY-BODY

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NODAL PROPERTIES OF FERMION WAVE FUNCTIONS L. MITAS∗ and M. BAJDICH Center for High Performance Simulation and Department of Physics, North Carolina State University, Raleigh, NC 27695, USA ∗ E-mail: lubos [email protected] www.ncsu.edu We briefly introduce some of the properties of nodal hypersurfaces and topologies of eigenstates of Hermitian operators. We focus on fermion nodes and illustrate their key features on a few-electron exactly solvable examples. In particular, we mention the property of generic fermionic ground states, which exhibit the minimal number of two nodal cells under rather general conditions as we have shown recently. We illustrate the impact of interactions on nodal topologies and we compare properties of Hartree–Fock and pairing pfaffian wave functions. A straightforward analysis of small solvable systems reveals that unlike Hartree–Fock, the pfaffian wave functions have the correct nodal topologies for both spin-polarized and unpolarized cases. Keywords: Fermion nodes; pfaffians; quantum Monte Carlo.

1. Introduction Let us consider a quantum system of N fermions in D−dimensional space in a state Ψ(r1 , ..., rN ) = Ψ(R). We assume the system Hamiltonian is time reversal invariant and without spin interactions so that Ψ(R) is real. The fermion node of Ψ(R) is a subset of configurations for which the wave function vanishes because of antisymmetry and it can be defined implicitly as Ω = {R; Ψ(R) = 0}. Although the wave function can vanish also due to other reasons such boundary conditions, imposed symmetries, singular interactions, etc, we assume that such zero sets are excluded from Ω. The fermion node divides the space of configurations into domains (cells) so that the nodal domain is defined as a simply connected subspace of configurations for which the wave function sign is constant (+ or −). The interest in nodes of eigenstates of Hermitian operators goes back to Hilbert and Courant. In particular, they studied nodes of 2D and 3D eigenstates of single-particle problems and discovered some of the properties of eigenstates nodal patterns (e.g., that an n-th eigenstate has n or less nodal domains). A number of studies over the past decades further explored several lines of related research such as statistical properties of nodes for n → ∞, relationships of nodes vs. types of spectra, volume of nodes for large n, connection to chaos, etc. The importance of nodes has been early recognized in quantum Monte Carlo

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(QMC) methods.1 It is well-known that due to the sign problem, QMC simulations of fermion systems are very inefficient. In general, QMC evaluations of fermionic averages scale exponentially in computer time with the number of particles N . If the exact node is known, however, QMC enables us to reformulate the original fermionic problem into an equivalent bosonic one and to evaluate exact energies and other quantities in computer time which scales as low polynomial in N . In this way the antisymmetry is mapped onto a boundary condition which is very easy to implement in the QMC methods. Even with approximate nodes the so-called fixed-node QMC has been very successful in real applications and in many cases has become the method of choice for large systems.1 However, the fixed-node bias can affect the QMC results and better understanding of fermion nodes could lead to a significant progress in solving important quantum many-body problems. The focus of this short paper is to briefly introduce some of the properties related to the nodes of eigenstates and illustrate the nodal properties of fermionic ground states on a few analytically solvable cases.

2. Nodes of Eigenstates The eigenstates nodes have been investigated mainly in low-dimensional cases for single-particle Hamiltonians and a number of properties have been identified. Most of them are still rather weak to provide detailed information about many-body fermion nodes nevertheless they reveal connections to other phenomena or physical quantities. Consider Hamiltonians with potentials which are not too singular (i.e., singularities are integrable and of zero measure). Let us list some of the most relevant mathematical features known (see, for example, Ref. 2 and references therein): (a) in a generic case the nodal surface is a manifold (generic means almost always and without additional constraints such as, e.g., imposed symmetries); (b) for the n-th excited state the number of nodal domains is n or less; (c) if two nodal hypersurfaces cross then the crossing angle is π/2, in case of k crossing hypersurfaces the crossing angles are all π/k; (d) volumes of nodes (Hausdorff measure) are bounded from both above and below with bounds proportional to powers of n. One of the complications is the fact that the node is a manifold generically, i.e., applies almost always. However, in quantum many-body problems we can often encounter non-generic cases, for example, imposing symmetries can make the problem non-generic and cause crossings of nodal hypersurfaces. The simplest example is the spin singlet (both spin channels occupied by the same number of particles) with a noninteracting Hamiltonian. The two spin channels and associated antisymmetries imposed on the state make the case non-generic. Another non-generic case are fermions with harmonic interactions which can be separated into spin-up and -down noninteracting subspaces.

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Recently, some progress has been made in understanding the nodes. A new exact node for a three-fermion ground state system has been discovered3 and a rather general theorem for nodes of noninteracting spin-polarized systems has been formulated.4,5 We were able to demonstrate that closed-shell non-degenerate fermionic ground states have only two nodal cells (i.e., one + and one −) for any size under rather general conditions. In addition, we have shown that interactions lead to the same property also for spin singlets by exploring the variational freedom of pairing wave functions. The two nodal domains property is intuitively appealing since the node is energetically costly and therefore Nature maximizes the volume of nodal domains. The configuration space for ground state fermions is thus bisected. Unfortunately, very little is known about exact nodes for realistic systems. There are a few cases where the exact nodes were found for two or three-particle systems with special symmetries. 2.1. He triplets The first case discovered some time ago is the node of two electrons in He atom in the lowest triplet state 3 S(1s2s). The node is determined solely by symmetry and is independent of interaction provided it is not too singular. Because of the spherical symmetry the wave function will depend only on three distances Ψ(r1 , r2 , r12 ). The exchange of the two particles then implies that the node is given by the condition r1 = r2 or, equivalently, r12 = r22 . The last condition then shows that the node is a quadratic manifold, basically, a 5D hyperboloid in 6D space (for a reminder, 2D hyperboloid in 3D is given as z 2 ± const = x2 − y 2 ). Another known two-electron He state is the lowest triplet of P symmetry and even parity 3 P (2p2 ). Assuming z is the P symmetry axis the node is given by the condition z0 · (r1 × r2 ) = 0 where z0 is the unit vector in z direction. This shows that the node positions are such that the two electrons lie on a plane which contains the z−axis.6 Finally, one can show that all the excited states for both symmetry sectors 3 S and 3 P are such that they contain the ground state fermionic node. Additional nodes which are present in the excited states have therefore bosonic character. 2.2. Three-electron quartet Remarkably, another case of exact node has been discovered for three electrons in the Coulomb potential. Consider three spin-polarized electrons in an S state with odd parity. The lowest energy state for this combinations of symmetries is the quartet 4 S(2p3 ). The state has a node given by an exact condition r1 · (r2 × r3 ) = 0 which, similarly to the previous cases, is independent of interactions. The combination of symmetries is so restrictive that interactions leave the node invariant and this is in fact the case also for the excited states. Therefore all the excited states have the property that they all contain the ground state fermionic node. This can be shown by rotating the one of the electrons to be on the z axis and decreasing the

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symmetry from spherical to cylindric. This changes the problem from three to two electrons in a P state with fixed axis, i.e., the preceding case of He P triplet. The same property that all excited state contain the ground state fermionic node thus immediately applies for this case as well. This has also implications that the oneparticle s symmetry channel is absent from the whole spectrum in this symmetry sector. One can show this as follows. We expand the exact states in this symmetry sector in excitations given as linear combinations of determinants of one-particle orbitals X |4 S(p3 )i = cijk det[|ni li mi i|nj lj mj i|nk lk mk i], (1) i,j,k

where the sum is over the complete one-particle spectrum (including scattering states). These states are such that coefficients cijk vanish for any excitation which contains l = 0 orbitals. Consider first states with one of the single-particle states to be of the s type, say, li = 0. That implies that all determinants in this configuration must have lj = lk otherwise the Clebsch–Gordan coefficients for S symmetry vanish. However, then the excitation would be of even parity and therefore such configurations drop off. Configurations with li = lj = 0 imply that also lk = 0 but then the parity is wrong again. Therefore the one-particle channel with s symmetry is “switched-off” completely. This property was used when showing that the fixednode calculation of this state provided the exact energy even when using nonlocal pseudopotential with s nonlocality.3 3. Two Nodal Cells of Ground States Note that in the previous cases of exact nodes we found that the ground states have the following important property: there were only two nodal cells. For example, in the triplet S case the two nodal domains corresponded to r1 > r2 and r1 < r2 . In the S quartet the two domains are specified by the left- or right-handedness of the product r1 · (r2 × r3 ). The topology is important for QMC calculations since it determines the sampling of the configuration space by the QMC stochastic processes (evolution of random walkers ensemble). A Typical feature of mean-field wave functions is that they divide the space into too many nodal domains, i.e., the space of configurations is too compartmentalized. For example, Hartree–Fock (HF) wave functions of spin-unpolarized systems have at least four (or, possibly, more) nodal cells since the node of ΨHF = Ψ↑ Ψ↓ is the direct product of the nodal surfaces of spin-up and -down subspaces and thus the nodal count is at least 2 × 2 = 4. Some time ago Ceperley7 conjectured that the fermionic ground states have only two nodal cells in generic cases. He was able to demonstrate this for a few hundred noninteracting periodic spin-polarized gas particles in 2D and 3D using a numerical proof. The proof was based on the following two properties: (a) Non-degenerate ground state wave functions satisfy the so-called tiling property which states that by applying all possible particle permutations to an arbitrary

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nodal cell one covers the entire configuration space. Unfortunately, this does not specify how many nodal cells are present, only gives the upper bound (N !). This upper bound is, in fact, saturated for 1D systems. (b) Let us consider three particles i, j, k from a spin-polarized system described by wave function Ψ(R). We call the particles i, j, k connected, if there exists a triple exchange ijk → jki path that does not cross the node, i.e., |Ψ(R)| > 0 along the exchange path. More connected particles can form a connected cluster of exchanges. If there exists a point Rt such that triple exchanges connect all the particles into a single connected cluster then Ψ(R) has only two nodal cells. This is easy to understand once we realize that the connected cluster of triple exchanges enables to exhaust all permutations of a given parity (say, even) and the tiling property then implies that there is no space left: there is only one cell for all even permutations and one cell for all odd permutations. Recently, we were able to explicitly demonstrate the two nodal cells for a large class of noninteracting spin polarized systems (harmonic fermions, fermions a sphere surface, periodic gas) for any size and D > 1.4,5 As we mentioned above for spin polarized systems the simplest Hartree–Fock wave functions have at least four nodal cells. Along the same lines we were then able to show that for interacting singlets the correlated wave functions (see below) smooth-out the multiple nodal cells into the minimal two.4,5,8

3.1. Pairing wave functions In most applications, the quantum Monte Carlo wave functions are of the JastrowSlater type, where the antisymmetric Slater part has the advantage of easy construction using one-particle orbitals from the best available one-particle theories.1 From the mentioned results it has become obvious that this form is not general enough and that a single product of spin-up and -down determinants typically produces an incorrect topology. Although in many cases the error is not large it is clear that in general one would like to start from a functional form which is able to describe the two domain topology of the ground state node correctly. Although one can always add more determinants for large systems, such approach leads to non-compact wave functions and rapidly growing computational time for large systems. The simplest generalization of one-particle orbitals are two-particle or pair orbitals which are able to capture pair correlations at the orbital level. In particular, the Bardeen–Cooper– Schrieffer (BCS) wave function, which is an antisymmetrized product of singlet pairs, has been recently used to calculate atoms and molecules9 as well as quantum liquids.10 Consider 1, 2, ..., N spin-up and N + 1, ..., 2N spin-down electrons in a singlet state. An antisymmetrized product of singlet pair orbitals φ(i, j) = φ(j, i) is the BCS wave function ΨBCS = A[φ(i, j)] = det[φ(i, j)] = det[Φ],

(2)

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which is simply a determinant of an N × N matrix. The BCS wave function is efficient for describing systems with single-band correlations such as Cooper pairs in conventional BCS superconductors where pairs form one-particle states close to the Fermi level. For partially spin-polarized states one can augment the matrix by columns/rows of one-particle orbitals, however, the spin-polarized subspace is then uncorrelated. For a fully spin-polarized system one ends up with the usual Hartree–Fock wave function and later we will show that this is inadequate. HF can produce wrong nodal surfaces even in the spin-polarized cases. In order to correlate spin-polarized electrons it is necessary to generalize the wave function form and to introduce effects of triplet pairing. For a system of 2N fully spin-polarized electrons the pairing wave function is formed as an antisymmetrized product of triplet pair orbitals χ(i, j) = −χ(j, i) and is given by A[χ(1, 2)χ(3, 4)...] = pf[χ(i, j)] = pf[χ] what defines a pfaffian of degree 2N . For example, for N = 2 we have   0 χ(1, 2) χ(1, 3) χ(1, 4) χ(2, 1) 0 χ(2, 3) χ(2, 4)  pf[χ(i, j)] = pf  χ(3, 1) χ(3, 2) 0 χ(3, 4) χ(4, 1) χ(4, 2) χ(4, 3) 0

= χ(1, 2)χ(3, 4) − χ(1, 3)χ(2, 4) + χ(1, 4)χ(2, 3).

(3)

(4)

Any pfaffian of an odd degree vanishes, however, the pfaffian wave function can be easily generalized to an odd number of electrons by extending the pfaffian by a row/column of one-particle (unpaired) orbital. For example, for three spin-up electrons, replace the last row/column in the equations above as χ(i, 4) → ϕ(i) and χ(4, i) → −ϕ(i). We note that the Hartree–Fock wave function is a special case of both BCS singlet and pfaffian triplet pairing wave functions. For example, we P HF HF recover HF from BCS by writing φ(i, j) = k ϕHF k (i)ϕk (j) where {ϕk (i)} are the occupied HF orbitals. The square of the pfaffian is related to the determinant of a skew-symmetric matrix as 2 pf[χ(i, j)] = det[χ(i, j)]. (5) However, in QMC applications one needs to calculate also the wave function sign, e.g., for enforcing the fixed-node boundaries. Although the sign can be traced along a continuous path of MC random walker it is much more efficient to evaluate the pfaffian directly. We have found a way how to calculate pfaffians by an O(N 3 ) algorithm which is analogous to the Gauss elimination for determinants.11 The algorithm is based on similarities in elementary operations for pfaffians and determinants. It is straightforward to show that pfaffian can be expanded in pfaffian minors and by exploring original Cayley’s analysis of pfaffian properties one can calculate the pfaffian and its updates for electron moves in computer time similar to the calculation of determinants. Let us now consider a partially spin-polarized

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system with unpaired electrons. The algorithm to calculate the pfaffian enabled us to propose a new wave function with singlet/triplet/unpaired orbitals defined as ΨP F = pf[φ(i, j), χ(k, l), ϕ(m)]  ↑↑  χ Φ↑↓ ϕ↑ = pf −Φ↑↓T χ↓↓ ϕ↓ 

(6)

−ϕ↑T −ϕ↓T 0

where the bold symbols are block matrices/vectors of corresponding pair orbitals and T denotes transposition. For a spin-restricted wave function the pair and oneparticle orbitals of spin-up and -down channels would be identical with further details given elsewhere.11 We just note that this is the first application of pfaffians to real electronic structure problems. 3.2. Be atom and the BCS wave functions In some simple cases with high symmetries it is easy to demonstrate that the pfaffian wave functions have only two nodal domains. Let us show this for the Be atom, which is well understood from the study of Bressanini and coworkers.12 In order to simplify the analysis we set the triplet pairings to zero so that the pfaffian simplifies to the BCS determinant. Consider the singlet pairing orbital of the type | ↑↓i = | ↑↓iHF + α| ↑↓icorr ,

(7)

where α is an expansion coefficient. We can write | ↑↓iHF = |φ1s φ1s i + |φ2s φ2s i

(8)

while the dominant correlation is captured by | ↑↓>corr = |φ2px φ2px i + |φ2py φ2py i + |φ2pz φ2pz i.

(9)

Clearly, the pairing function has the required S symmetry and it is straightforward to show that the wave function has only two nodal cells. Configure the two spin-up electrons at the antipodal points on a sphere with radius ra > 0. In the plane defined by the spin-up electrons and the origin, place the spin-down electrons at antipodal points of a circle with different radius, say, rb > 0 and rb 6= ra . In this configuration the Hartree–Fock wave function vanishes (particles lie on the Hartree–Fock node). In general, however, the BCS wave function does not vanish since we have det[hri , rj | ↑↓i] = C r21 · r43

(10)

where C is a symmetric nonzero factor. We can now rotate all four particles in the common plane by π and the wave function remains constant. The rotation exchanges the two electrons in the spin-up channel and the two electrons in the spin-down channel showing that the spin-up and -down nodal domains are connected (otherwise we would encountered zeros due to node crossing in both spin-channels).

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3.3.

4

S(1s2s3s) state and the pfaffian wave function

As we have mentioned elsewhere,4 imposing symmetries can lead to more than the minimal two cells. For example, the lowest atomic quartet of S symmetry and even parity is 4 S(1s2s3s). In the noninteracting limit this state exhibits six nodal cells. The reason is that all the single-particle orbitals depend only on radii and therefore the nodes have effectively one-dimensional character. It is interesting that even in this case the interaction does change the topology and the number of nodal cells decreases to minimal two. In the original paper we showed this numerically for the Coulomb potential. For simplicity, let us illustrate this explicitly on a simpler model of harmonically confined electrons while using the same symmetries of states and one-particle orbitals as for the Coulomb case (with more work and space one can do the explicit proof even for the Coulomb potential). The harmonic oscillator has the advantage that the exponential Gaussian prefactor is the same for all states and one-particle orbitals irregardless of the subshell or occupations. We conveniently omit these Gaussian prefactors in all expressions since they are irrelevant for the nodes. The noninteracting (Hartree–Fock) wave function then becomes simply

ΨHF (r1 , r2 , r3 ) = (r22 − r12 )(r32 − r12 )(r32 − r22 )

(11)

since φ2s and φ3s are first and second order polynomials in r 2 , respectively. It is clear that there are six nodal domains, because ΨHF vanishes whenever ri = rj and there are six permutations of r1 , r2 , r3 orderings. Now, we can switch on the interactions and describe the resulting correlations by expanding the wave function in excitations. The excitations with higher s symmetry orbitals do not change the nodal structure and their contributions to the exact wave function are actually very small. The dominant excited configuration is 4 S(1s, 2p, 3p), which introduces angular correlations. We will show that this effect is described by the pfaffian with an appropriate triplet pairing orbital (the singlet pairing vanishes for spin-polarized systems by definition). Let us position the particles into a configuration which would simplify the expressions as much as possible. We then demonstrate that it is possible to perform a triple exchange without crossing the node, verifying thus the two nodal cell property. For this purpose we position particles as follows: r1 = [0, ra , 0], r2 = [0, 0, 0] and r3 = [rb , 0, 0] where ra > rb > 0. During the exchange the particles first move as follows: 1 → 2, 2 → 3 and 3→ [ra , 0, 0]. The exchange is then finished by rotating particle 3 by +π/2 around the z−axis to the position [0, ra , 0], completing thus the repositioning 3 → 1. The particle 1 moves to the origin along the y−axis while particles 2, 3 move along the x−axis while their radii are such that r3 > r2 is maintained during the exchange. Clearly, during the exchange the node of noninteracting wave function will be crossed twice. This will happen for the following two configurations: when r1 = r3 and again when r1 = r2 . We will now show that for the pfaffian wave function the exchange can be carried out without

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crossing the node. We write the triplet pairing function as χ(i, j) = φ2s (i)φ3s (j) − φ2s (j)φ3s (i)

+ β[φ2px (i)φ3px (j) − φ2px (j)φ3px (i) + (x → y) + (x → z)],

(12)

where i, j = 1, 2, 3 and β is the expansion coefficient. Since we have only three particles the ”unpaired” column/row in the pfaffian of the 4×4 matrix is such that χ(i, 4) = φ1s (i). Using the well-known forms of φ2px = x and φ3px = x(r2 − 1) and using the formula for pfaffian above we can explicitly find that pf[χ(i, j)] = (r32 − r22 )[(r22 − r12 )(r32 − r12 ) + βx2 x3 ]

(13)

and assume that β > 0. Typical value of β is of the order of 10−2 showing that triplet correlations are quite small. (In the case β < 0, it is straightforward to modify the positions and exchange path accordingly: the initial position of the particle 3 should be [−rb , 0, 0], it moves to [−ra , 0, 0] and it is then rotated by −π/2 around the zaxis.) After a little bit of thinking one finds that this function does not change the sign provided the difference between the radii r2 and r3 when r1 = (r2 + r3 )/2 is sufficiently small so that the following inequality holds |r2 − r3 + O[(r3 − r2 )2 ]| < β,

(14)

which is easy to guarantee with some care while moving the particles along the exchange path. The minimum of the radial part appears essentially at the point r1 = (r2 + r3 )/2 and the value is negative. However, the other term keeps the wave function positive providing the minimum is not too deep. Since the minimum can be made arbitrarily shallow by adjusting radii r2 , r3 to be sufficiently close to each other one can easily fulfill the inequality. The wave function then remains positive along the whole exchange path. This exercise is not trivial: for a given spin polarized state we have explicitly found that the pfaffian wave function with triplet pairing has a different topological structure than the Hartree–Fock wave function. Although the correlation effect in triplet pairing is usually very small, it is nonzero. The demonstrated change in topology is indeed quite interesting and not obvious since in this case the six nodal cells fuse to the minimal two as a result of correlation. This is perhaps the simplest illustration of the fact that pfaffians have important variational freedom beyond what is present in the BCS wave function with only singlet pairing. 4. Summary In summary, we have briefly introduced the notion of eigenstate nodes and we explained some of the important aspects of the fermion nodes for QMC methods. We have illustrated cases of a few exact nodes known for two and three electrons in central potentials. Finally, we have shown that the pfaffian wave functions have a remarkable variational freedom which enables us to capture the impact of electron correlation on nodal topologies for both spin-polarized and unpolarized states.

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Acknowledgment This research has been supported by NSF DMR-0121361 and EAR-0530110 grants. Collaboration with K. E. Schmidt is gratefully acknowledged. References 1. W. M. C. Foulkes, L. Mitas, R. J. Needs and G. Rajagopal, Rev. Mod. Phys. 73, 33 (2001). 2. M. Caffarel, X. Krokidis and C. Mijoule, Europhys. Lett. 20, p. 581 (1992). 3. M. Bajdich, L. Mitas, G. Drobny and L. K. Wagner, Phys. Rev. B (Condensed Matter and Materials Physics) 72, p. 075131 (2005). 4. L. Mitas, Phys. Rev. Lett. 96, p. 240402 Jun. (2006). 5. L. Mitas, in review; cond-mat/0605550 (2006), submitted to Phys. Rev. B. 6. D. Bressanini and P. J. Reynolds, Physical Review Letters 95, p. 110201 (2005). 7. D. M. Ceperley, J. Stat. Phys. 63, p. 1237 (1991). 8. M. Bajdich, L. Mitas, G. Drobn´ y, L. K. Wagner and K. E. Schmidt, Phys. Rev. Lett. 96, p. 130201(Apr 2006). 9. M. Casula and S. Sorella, The Journal of Chemical Physics 119, 6500 (2003). 10. J. Carlson, S.-Y. Chang, V. R. Pandharipande and K. E. Schmidt, Phys. Rev. Lett. 91, p. 050401 Jul. (2003). 11. M. Bajdich, L. Mitas, L. K. Wagner and K. E. Schmidt, cond-mat/0610850 (2006), submitted to Phys. Rev. B. 12. D. Bressanini, D. M. Ceperley and P. J. Reynolds, in Recent Advances in Quantum Monte Carlo Methods II, Ed. W. A. Lester, S. M. Rothstein, and S. Tanaka, World Scientific, Singapore (2002) 119, p. 6500 (2002).

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SIMULATING ROTATING BEC: VORTICES FORMATION AND OVER-CRITICAL ROTATIONS Siu A. Chin Department of Physics, Texas A&M University, College Station, TX 77843, USA By exactly solving the rotating anisotropic harmonic oscillator, we derive a class of very efficient second and fourth order algorithms for evolving the Gross–Pitaevskii equation in imaginary time. The resulting algorithms are used to study spontaneous vortices nucleations and over-critical rotations. Keywords: Rotating BEC; Gross–Pitaevskii equation; vortices nucleation; giant vortex.

1. Introduction It has been known for some time that the first order pseudo-spectral, split-operator method1 is a very fast way of solving the non-linear Schr¨ odinger equation. However, first or second order split operator methods2,3 and Crank-Nickolson (CN) algorithms with4 or without5 splitting ignore the time-dependence of the non-linear potential and converge linearly or quadratically only at very small time steps. Bandaruk and Shen6 have applied higher order decomposition schemes with negative coefficients to solve the real time non-linear Schr¨ odinger equation. These negative time step algorithms cannot be used for imaginary time evolution because the kinetic energy propagator is the diffusion kernel and diffusion is time-irreversible.7–10 Recently we have derived a class of very accurate fourth order factorization algorithm11 for solving the GP equation in imaginary time. These algorithms solves the rotating anisotropic harmonic oscillator exactly and uses forward, all positive time step algorithms12–17 to achieve fourth order accuracy. Here we use them to study the spontaneous vortices nucleation18,19 and giant vortex formation20 in overcritical rotations. 2. Solving the Rotating Anisotropic Harmonic Oscillator First, we will show how the rotating anisotropic harmonic oscillator H=

1 2 2 1 2 2 ˜ 1 2 (p + p2y ) + ω ˜ x + ω ˜ y − Ω (xpy − ypx ) , 2 x 2 x 2 y

(1)

can be solved exactly as a splitting algorithm. Hamiltonian (1) is well-studied in nuclear physics21 and can be exactly decomposed into two independent harmonic

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oscillators as follows. It is convenient to characterize22 the anisotropy via the deformation parameter δ ω ˜ x2 = (1 + δ)ω02 ,

ω ˜ y2 = (1 − δ)ω02 , (2) √ measuring lengths in units of the oscillator length l = 1/ ω0 , and express H and ˜ in units of ω0 . The resulting dimensionless Hamiltonian is then Ω H=

1 2 1 1 (p + p2y ) + (1 + δ)x2 + (1 − δ)y 2 − Ω (xpy − ypx ) , 2 x 2 2

(3)

˜ 0 . To diagonalize this Hamiltonian, we introduce canonical transforwhere Ω = Ω/ω mations, Q1 = α1 (cx − spy ),

P1 =

1 (cpx + sy), α1

(4)

Q2 = α2 (cy − spx ),

P2 =

1 (cpy + sx), α2

(5)

where αi are normalization constants, and c = cos(φ), s = sin(φ). By fixing φ via tan(2φ) =

2Ω , δ

(6)

and αi to normalize the Pi2 terms with unit coefficient, the transformed Hamiltonian is then 1 1 1 1 (7) H = T1 + V1 + T2 + V2 = P12 + Ω21 Q21 + P22 + Ω22 Q22 , 2 2 2 2 with p Ω21 = 1 + Ω2 + δ 2 + 4Ω2 , p Ω22 = 1 + Ω2 − δ 2 + 4Ω2 . (8) √ 2 As Ω increases, Ω2 crosses zero and becomes negative at Ω = 1 − δ. At this critical rotation frequency, the centrifugal force overcomes the weaker harmonic potential in the y-direction and the particle is free in that direction. Eq. (7) consists of two independent harmonic oscillators with frequencies Ω1 and Ω2 . For a single harmonic oscillator, H =T +V =

1 2 1 2 2 p + ω x , 2 2

(9)

the corresponding imaginary time propagator can be exactly factorized 11 as e−τ (T +V ) = e−τ CE V e−τ CM T e−τ CE V = e−τ CE T e−τ CM V e−τ CE T ,

(10)

where the edge and middle coefficient functions CE and CM are given by CE =

cosh(ωτ ) − 1 ωτ sinh(ωτ )

and CM =

sinh(ωτ ) . ωτ

(11)

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To propagate (7) in imaginary time, each harmonic oscillator can be evolved via (10). However, since T1 and V2 only depend on px and y, they can be grouped next to each other so that both can be evaluated in the same mixed representation described below. Similarly, T2 and V1 only depend on x and py and can be grouped together. Thus we propagate (7) as e−τ (T1 +V1 +T2 +V2 ) = e−τ CE (1)T1 −τ CE (2)V2 e−τ CM (1)V1 −τ CM (2)T2 e−τ CE (1)T1 −τ CE (2)V2 . (12) Here we use the shorthand notations CE (1) = CE (Ω1 ), CM (2) = CM (Ω2 ), etc.. To implement (12), let Z 1 ψ(x, y) = √ dpx ψ(px , y) eipx x , (13) 2π Z 1 ψ(px , y) = √ dx ψ(x, y) e−ipx x , 2π and

Z 1 ψ(x, py ) = √ dy ψ(x, y) e−ipy y , 2π Z 1 dy dpx ψ(px , y) eipx x−ipy y . = 2π

(14)

The operators T1 and V2 are diagonal with respect to ψ(px , y) and T2 and V1 are diagonal with respect to ψ(x, py ). In practice, ψ(x, y) is discretized as an N × N complex array with Fourier transforms computed using the discretized FFT. The exact algorithm (12) corresponds to four steps: (1) Compute the forward N -1D transform ψ(px , y) from ψ(x, y) and multiply ψ(px , y) grid-point by grid-point by e−τ CE (1)T1 −τ CE (2)V2 , where T1 and V2 are now understood to be functions of px and y. (2) Compute the 2D transform ψ(x, py ) from the updated ψ(px , y) and multiply ψ(x, py ) by e−τ CM (1)V1 −τ CM (2)T2 , where V1 and T2 are now functions of x and py . (3) Compute the inverse 2D transform from the updated ψ(x, py ) back to ψ(px , y) and multiply ψ(px , y) by e−τ CE (1)T1 −τ CE (2)V2 . (4) Compute the backward N -1D transform from the updated ψ(px , y) back to ψ(x, y). 3. Solving the Gross–Pitaevskii Equation The 2D Gross–Pitaevskii equation with a rotating anisotropic trap is ( T + g|ψ|2 )ψ(x, y) = µψ(x, y).

(15)

where now T denotes the entire rotating trap Hamiltonian (7) T = T1 + V1 + T2 + V2 .

(16)

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The condensate ground state can be projected out by imaginary time evolution: ψ0 ∝ lim ψ(τ ) = lim e−τ [T +V (τ )]+τ µ ψ(0). τ →∞

τ →∞

(17)

The chemical potential µ is determined by preserving the wave function’s normalization to unity. This will always be assumed and this term will not be mentioned in the following discussion. Since ψ(τ ) is time-dependent, the Gross–Pitaevskii potential V (τ ) = g|ψ(τ )|2 ,

(18)

is also time-dependent. In this case, to solve (17) by factorization algorithms, one must apply rules of time-dependent factorization:14,23 the time-dependent potential must be evaluated at an intermediate time equal to the sum of time steps of all the T operators to its right. For example, the first order algorithm 1A is ψ(∆τ ) = e−∆τ T e−∆τ V (0) ψ(0)

(19)

and the first order algorithm 1B is ψ(∆τ ) = e−∆τ V (∆τ ) e−∆τ T ψ(0).

(20)

Algorithm 1A can be implemented in a straightforward manner. Algorithm 1B requires that the potential be determined from the to-be-computed wave function. This self-consistency condition seemed difficult to satisfy. However, in imaginary time the wave function converges quickly to an approximate but stationary ground state such that ψ(∆τ ) is essentially ψ(0). Thus after some initial iterations, the normalized g|ψ(∆τ )|2 is independent of τ and can be replaced by g|ψ(0)|2 . Thus at small ∆τ one can approximates g|ψ(∆τ )|2 by g|ψ(0)|2 in (20). We shall refer to this version of the algorithm as 1B0. The second order algorithm 2A can be defined as 1

1

ψ(∆τ ) = e− 2 ∆τ V (∆τ ) e−∆τ T e− 2 ∆τ V (0) ψ(0)

(21)

and algorithm 2B as 1

1

ψ(∆τ ) = e− 2 ∆τ T e−∆τ V (∆τ /2) e− 2 ∆τ T ψ(0).

(22)

Similarly, one can replace g|ψ(∆τ )|2 by g|ψ(0)|2 in algorithm 2A without affecting its quadratic convergence at very small ∆τ . We shall refer to this version of the algorithm as 2A0. Algorithm 2B was not implemented because it required two executions of the exact algorithm (12), which is less efficient. Figure 1 shows the convergence of algorithm 1A and 1B0 for the chemical potential µ. Both are clearly first order at small ∆τ . The calculation is done for δ = 0.5, Ω = 0.5, and g = 50. This choice corresponds to sizable anisotropy, rotation, coupling strength and not close to any particular limit. The calculation uses 642 grid points over a 142 harmonic length square centered on the origin. (Changing the grid size to 1282 only changes the stable results in the fifth or sixth decimal place.) The ground state wave function is nearly converged by τ = 2. The chemical potential shown is calculated at τ = 10. Algorithm 2A0 is also clearly second order. Also shown are results obtained by enforcing the self-consistent condition, denoted by the suffix “w”. Interested readers can consult Ref. 11 for further details.

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Fig. 1. The convergence of first and second-order algorithms in computing the chemical potential of the Gross–Pitaevskii equation in a rotating anisotropic trap. The lines are fitted linear and quadratic curves to algorithm 1A, 1B0 and 2A0 to demonstrate their respective orders of convergence.

4. Forward Fourth-order Algorithms Splitting algorithms such as (19) to (22) can be generalized to higher order via 24–31 e−∆τ (T +V ) =

Y

e−ai ∆τ T e−bi ∆τ V ,

(23)

i

with suitable coefficients {ai , bi }. However, as first proved by Sheng32 and Suzuki,33 beyond second order, any factorization of the form (23) must contain some negative coefficients in the set {ai , bi }. Goldman and Kaper34 later proved that any factorization of the form (23) must contain at least one negative coefficient for both operators. Since the kinetic energy propagator in imaginary time is the diffusion kernel, negative time steps cannot be allowed because diffusion is time-irreversible. Thus none of the traditional higher factorization schemes can be used to evolve the imaginary time Schr¨ odinger equation or the Gross–Pitaevskii equation. To go beyond second order, one must use forward factorization schemes with only positive factorization coefficients.12–17 Since T is computational demanding, we must choose a fourth order algorithm with a minimal number of T operators. Thus among the many forward algorithms discovered so far,13–17 we choose to implement only the simplest algorithm, 4A. 1

1

2

e

1

1

ψ(∆τ ) = e− 6 ∆τ V (∆τ ) e− 2 ∆τ T e− 3 ∆τ V (∆τ /2) e− 2 ∆τ T e− 6 ∆τ V (0) ψ(0) ,

(24)

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Fig. 2. The convergence of fourth-order algorithm 4A00 (solid diamonds) in computing the chemical potential of the Gross–Pitaevskii equation in a rotating trap as compared to first and second order algorithms. Algorithm 4A0W (circles) and 4AWW (solid circles) are further improved versions which enforce the self-consistency condition.

with Ve given by

∆τ 2 Ve = V + [V, [T, V ]] . (25) 48 Despite the seeming complexity of T as defined by the Hamiltonian (7), we have remarkably, 2 2 ∂V ∂V + . (26) [V, [T, V ]] = ∂x ∂y Thus the midpoint effective potential is ∆τ 2 g 2 ∂ |ψ(∆τ /2)|2 2 ∂ |ψ(∆τ /2)|2 2 2 e . (27) V (∆τ /2) = g|ψ(∆τ /2)| + + 48 ∂x ∂y

The partial derivatives can be computed numerically 9 using FFT. To implement this fourth order algorithm, we can replace ψ(∆τ /2) and ψ(∆τ ) by ψ(0). We will refer to this as algorithm 4A00. Its convergence is shown in Fig. 2. We have retained some first and second order results for comparison. Aside from its abrupt instability at ∆τ ≈ 0.3, its convergence is remarkably flat. All the results at ∆τ < 0.3 differ only in the fifth decimal place. For other implementations shown involving the self-consistency condition, see Ref. 11. The computational effort required by each algorithm is essentially that of evaluating the exact algorithm (12), which uses 3 2D-FFT. Since 1A, 1B0, and 2A0 all

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6 4 2 y

0 -2 -4 -6 -6

-4

-2

0

2

4

6

x Fig. 3. Vortices are nucleated from the edge of the rotating trap and move toward the center. Here Ω = 0.6 is sufficiently high to nucleate two vortices.

use the exact algorithm once, the second order algorithm 2A0 is clearly superior. Algorithms 4A00 requires two evaluations of the exact algorithm plus the gradient potential. The gradient potential, if done by FFT, requires 2 2D-FFT. Thus algorithm 4A00 requires 8 2D-FFT, which is 8/3 ≈ 3 times the effort of algorithm 2A0. Since algorithm 4A00 converges much better than 2A0 at time-steps more than three times as large, algorithm 4A00 is clearly more efficient than second order algorithms. For example, the GP ground state can be obtained at τ = 2. Using algorithm 4A00 at ∆τ = 0.3, one can get there in 7 iterations. Algorithm 2A0 would have taken 80 iterations at ∆τ ≈ 0.02 . By solving the rotating anisotropic harmonic oscillator (12) exactly, we have effected a tremendous simplification allowing us to derive very compact fourthorder algorithms with excellent large time-step convergences. If we did not have the 1 exact propagator, then we would have to approximate each occurrence of e − 2 ∆τ T in (24) to fourth order, resulting in a much more complex algorithm. 5. Vortex Dynamics The imaginary time propagation of the GP equation can be used to model the dissipative process of vortex formation18,19 in a Bose–Einstein condensate. During the oral presentation of this work, real time simulations were ran to show how rotation in an anisotropic trap spontaneously nucleates vortices from the edge of the trap. For a fast real time demonstration, only the second order algorithm 2A0 is used. For demonstration purposes, the qualitative picture of vortices formation

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6 4 2 y

0 -2 -4 -6 -10 -8 -6 -4 -2

0

2

4

6

8

10

x Fig. 4. At Ω = 0.95, 26 vortices are seen. To magnify the detail structure, the vertical scale is expanded, masking the fact that this fast rotating condensate is actually very elongated.

(e.g., the number of vortices at a given Ω) is unchanged even when the time-step size is fairly large. For the following discussion, we use N = 1282 , g = 100, δ = −0.05 and ∆t = 0.2 − 0.3. For a given trap configuration, there is a Ωmin below which no vortex can nucleate. Figure 3 shows the nucleation of two vortices when Ω = 0.6, above Ωmin . It is difficult to determine Ωmin empirically because the latency time for vortex nucleation is very long when Ω ≈ Ωmin . It is also difficult to nucleate an odd number of vortices. This is probably due to the symmetric way in which the algorithm is executed. When Ω = 0.95, which is close to the critical angular p frequency of Ωc = 1 − |δ| ≈ 0.97, the condensate elongates in the x-direction and nucleates many vortices, forming an array. This is shown in Fig. 4. The array is not very hexagonal, this may just be due to the small size of the simulation square. As Ω → Ωc , the condensate is no longer confined in the x-direction. To study overcritical rotation,20 we add to the harmonic trap of (3) a quartic potential 1 1 1 (1 + δ)x2 + (1 − δ)y 2 + λ(x2 + y 2 )2 . (28) 2 2 2 Fig. 5 shows the result for λ = 0.01 at Ω = 1.25, when the initial Gaussian condensate function is seeded with four unit of angular momentum (i.e., the initial Gaussian wave function is multiplied by (x + iy)4 ). For under-critical rotations, this four unit of angular momenta would fragment into four vortices with one unit of angular momentum each. Here in Fig. 5, after equilibrated to the ground state, one finds a giant vortex with multiple units of angular momentum at the center, surrounded by half-rings of unit angular momentum vortices along the weak (xdirection) side of the trap. Vtrap (x, y) =

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8 6 4 2 y

0 -2 -4 -6 -8 -8

-6

-4

-2

0

2

4

6

8

x Fig. 5. Overcritical rotation at Ω = 1.25. The condensate has a giant vortex at the center surrounded by a half-ring of vortices at both sides.

6. Concluding Summary In this work, by exactly solving the rotating anisotropic trap, we have derived a number of second and fourth order algorithms for solving the GP equation. The fourth order algorithms derived using forward factorization schemes, are the only splitting algorithms currently possible for solving the imaginary time GP equation. In contrast to other algorithms, generalizing these algorithms to 3D is very transparent, one simply replaces 2D-FFT everywhere by 3D-FFT. We have also demonstrated the efficiency of these algorithms in simulating the spontaneous nucleation of vortex arrays and the the formation of a giant vortex in over-critical rotations. Acknowledgments This work was supported in part, by a National Science Foundation grant No. DMS-0310580. References 1. 2. 3. 4. 5.

T. R. Taha and M. J. Ablowitz, J. Com. Phys. 55, p. 203 (1984). B. Jackson, J. F. McCann and C. S. Adams, J. Phys. B 31, p. 4489 (1998). C. M. Dion and E. Cances, Phys. Rev. E 67, p. 046706 (2003). S. K. Adhikari and P. Muruganandam, J. Phys. B 35, p. 2831 (2002). P. A. Ruprecht, M. J. Holland, K. Burnett, and M. Edwards, Phys. Rev. A 51, p. 4704 (1995).

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6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34.

A. D. Bandrauk and H. Shen, J. Phys. A 27, p. 7147 (1994). H. A. Forbert and S. A. Chin, Phys. Rev. E 63, p. 016703 (2001). H. A. Forbert and S. A. Chin, Phys. Rev. B 63, p. 144518 (2001). J. Auer, E. Krotscheck, and S. A. Chin, J. Chem. Phys. 115, p. 6841 (2001). O. Ciftja and S. A. Chin, Phys. Rev. B 68, p. 134510 (2003). S. A. Chin and E. Krotscheck, Phys. Rev. E 72, p. 036705 (2005). M. Suzuki, Computer Simulation Studies in Condensed Matter Physics VIII, Eds, D. Landau, K. Mon and H. Shuttler (Springler, Berlin, 1996). S. A. Chin, Phys. Lett. A 226, p. 344 (1997). S. A. Chin and C. R. Chen, J. Chem. Phys. 117, p. 1409 (2002). S. A. Chin and C. R. Chen, Cele. Mech. Dyn. Astron. 91, p. 301 (2005) S. A. Chin, Phys. Rev. E, 73, p. 026705 (2006). S. A. Chin, Phys. Lett. A, 354, 373 (2006) F. Dalfovo, S. Giorgini, L. Pitaevskii and S. Stringari, Rev. Mod. Phys. 71, p. 463 (1999). A. Fetter and A. Svidzinsky, J. Phys.: Condens. Matter 13, p. R135 (2001). A. Fetter, B. Jackson and S. Stringari, Phys. Rev. A 71, p. 013605 (2005). P. Ring and P. Schuck, The Nuclear Many-Body Problem, P.133, Springer-Verlag, Berlin-NY (1980). ¨ Oktel, Phys. Rev. A 69, p. 023618 (2004). M.O. M. Suzuki, Proc. Japan Acad. 69, Ser. B, p. 161 (1993). E. Forest and R. D. Ruth, Physica D 43, p. 105 (1990). M. Creutz and A. Gocksch, Phys. Rev. Letts. 63, p. 9 (1989). H. Yoshida, Phys. Lett. A150, p. 262 (1990). H. Yoshida, Celest. Mech. 56 p. 27 (1993). R. I. McLachlan, SIAM J. Sci. Comput. 16, 151 (1995). M. Suzuki, Phys. Lett. A146, p. 319 (1990); 165, p. 387 (1992). R. I. McLachlan and G. R. W. Quispel, Acta Numerica, 11, p. 241 (2002). E. Hairer, C. Lubich, and G. Wanner, Geometric Numerical Integration, SpringerVerlag, Berlin-New York, 2002. Q. Sheng, IMA J. Numer. Anal. 9, p. 199 (1989). M. Suzuki, J. Math. Phys. 32, p. 400 (1991). D. Goldman and T. J. Kaper, SIAM J. Numer. Anal., 33, p. 349 (1996).

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POLARIZABILITY IN QUANTUM DOTS VIA CORRELATED QUANTUM MONTE CARLO L. COLLETTI Istituto Nazionale di Fisica Nucleare, Sezione di Padova, Gruppo Collegato di Trento Trento, 38050, Italy ∗ E-mail: [email protected] and Free University of Bozen-Bolzano Bolzano, 39100, Italy F. PEDERIVA and E. LIPPARINI Dipartimento di Fisica, Universit` a degli Studi di Trento, Trento, 38050, Italy C. J. UMRIGAR Theory Center and Laboratory of Atomic and Solid State Physics, Cornell University Ithaca, New York, 14853, USA In this paper we review the calculations of charge-density and spin-density polarizabilities in small quantum dots by using a correlated Monte Carlo scheme. In the limit of small external fields, knowledge of polarizability implies, thanks to the commonly used “sum rules”, prediction of the excitation energy of the dipole mode. The need of a numerical approach arises when spin-density polarizability is pursued, while the charge-density mode is analytically calculable as long as the confinement is maintained parabolic. Keywords: Quantum dots; dipole modes; correlation energy; spin-density polarizability; correlated quantum Monte Carlo.

1. A Confined, Two-dimensional Electron System Quantum dots (QD) are finite many-electron systems. As in atoms, electrons in QD are confined by a central potential in such a narrow space that quantum mechanical effects show up, leading the electrons to occupy discrete energy levels obeying Pauli’s principle and Hund’s rules.1,2 Single-particle and collective state spectra of QD have been extensively studied by several groups and by employing different methodologies, such as density functional and variational and diffusion Monte Carlo3,4 . Here we consider small dots whose electrons are arranged in closed-shell configurations of the harmonic oscillator spectrum; the limited number of electrons allows us to be confident on the validity of the quadratic approximation of the potential and to include only one Slater determinant in the trial wavefunction, since the total

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spin is zero and all the single-particle orbitals are occupied up to the Fermi energy. The linear size of such dots is of the order of 100 nm. The electron density is thus much lower than that of natural atoms, then enhancing the role of the correlation energy with respect to the kinetic one. We quantify here the effect of the correlation as renormalizing factor of the spin-density dipole mode with respect to the charge-density dipole mode, which is unaffected by the electron-electron interaction. 2. A Link between Dipole-mode Excitation Energy and Polarizability The backbone of our approach is the connection between polarizability α and excitation energy ω as provided by sum rules m1 , m−1 when certain physical conditions can be granted. This is the case when the external perturbation is characterized by small momentum transfer so that the density wave can be approximated by the ˆ = λ PN xi , which displaces each one of the N charge-density dipole operator D i=1 electrons in the direction of the x−axis proportionally to the strength λ of the field. m1 does not depend on the correlations and can be analytically derived. m−1 is related to the polarizability of the system. In fact, given the ground and excited states of the unperturbed system, |0 > and |n >, and the excitation spectrum ωn0 , m−1 is defined as X | < 0|D|n ˆ > |2 (1) m−1 = ωn0 n

whereas, taking the classical definition of polarizability as the average displacement from a reference position of a distribution of electric charges when an external field is applied normalized to the intensity of the field and considering first-order expansion of this perturbed state ψλ on the unperturbed states |0 > and |n > the polarizability reduces to P ˆ > |2 | < 0|D|n α = −2 n . (2) ωn0 One then concludes that α = −2m−1 and, since the average excitation energy can be expressed as ω = (m1 /m−1 )0.5 one obtains ω = (−2m1 /α)0.5 (m1 is model independent). If we can calculate the polarizability we obtain also an estimate of the excitation energy for the dipole mode. There is nothing about many-body physics up to this point. In fact in such parabolically-confined dots, the dipole mode is a one-body excitation (plasmon), which is completely independent of the correlations among the electrons.5 Predicting and measuring the plasmon yields then no information on the many-body correlations within the dot. This information can be obtained looking at spin-density dipole excitation. Exciting the dot by the external field, still in the long-wavelength limit but with the addition of a spinorial component, results in a different excitation for spin-up and spin-down electrons. For this ˆ σ = λ PN σ z xi , everything we write above for the chargeexcitation operator, D i=1 i density dipole operator can be repeated, with the fundamental difference that the spin-dependent operator does not commute with the electron-electron potential,

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impeding an analytical solution for the sum rule m−1 for this operator and thus impeding an analytical estimate of the spin-density dipole excitation energy. This is where the need for a numerical solution arises. Our challenge is that of using Monte Carlo (MC) algorithms, known for leading to benchmark accuracy’s performances but whose application is traditionally limited to ground-state properties.

3. Quantum Monte Carlo Estimates of Polarizabilities A naive MC approach to polarizability would follow this iterative procedure: fix a value λ1 for the external field strength; accordingly, consider the perturbed system equivalent to an unperturbed system whose confinement has been rigidly shifted by a quantity proportional to λ1 ; sample configurations of the latter system with the usual MC scheme and accumulate the estimator for the (spin-)dipole operator over the configurations; after this first step of the procedure, an estimate of the polarizability could be given as α1 = λD1 ; recall that polarizability is given in terms of λ → 0; then a slightly different value of the external pertubation should be considered, λ2 , and, correspondingly, a fully new sampling of configurations carried on, yielding to α2 and so on; eventually, extrapolate to get polarizability in the λ → 0 limit. This approach would be computationally very time consuming. In fact each estimate αi should be obtained with a statistical error that has to be order of magnitudes smaller than the differences between the single αi values and this would imply long simulation times. An efficient alternative is that of a correlated sampling.6 In this approach, only sampling of a ”primary geometry”, with λ = 0, is required, while samples of ”secondary geometries”, with λ 6= 0, are simultaneously generated by shifting each particle in each configuration of the primary geometry by a fixed amount proportional to λ. Wavefunctions of the secondary geometries have the same shape as the wavefunction of the primary geometry, but with recentered coordinates, and are independently optimized. The estimate for polarizability is then accumulated over the sampled configurations, whose contributions are weighted with the ratio of the primary to the secondary wavefunction. In the diffusion MC approach the primary walk that projects the unperturbed ground state is generated according to the standard procedure, i.e. a population of configurations is evolved in imaginary time using an importance sampled approximate Green’s function of the Schr¨ odinger equation. The secondary walks, used to project out ψ λ , are generated again as described above. We must however take into account the different multiplicity of the primary and secondary walkers due to different Green’s functions that should be used, in principle, for the propagation. The bare shifting of the electrons could in fact be responsible for accepting moves that cross the nodes of the secondary wavefunctions, while not crossing those of the primary one. This problem has been addressed by imposing a further reweighting of the contributions by considering different time steps for the different geometries, regularly adapted on the fly for each secondary configuration, measuring the width of the diffusion step of the secondary sampling with respect to the primary one.

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4. Results and Comments We considered circularly symmetric external confinements and N=6,12,20. We expressed the results as the ratio of polarizabilites, thus as the relative ratio r of the charge-density mode energy with respect to the spin-density one. We found 1.165(1) < r < 1.85(2) by diffusion MC, while variational MC results show a very slow convergency. Diffusion MC runs proved anyway to be convergent with no or little dependence on the variational trial wavefunction used. If compared to other approaches, results indicate a better, though still qualitative, comparison with experimental results. In fact, from Ref. 7 one deduces r ' 2, while time-dependent local-density approximation calculations4 indicate r ' 3.5. Results are fully reported in Ref. 8. It appears clearly that the tendency of charge polarizability is that of decreasing when, keeping the number of electrons fixed, the strength of the confinement is increased. This has a simple interpretation as the electron density getting more rigid and less subject to distorsion when the confinement gets stricter. We found the same tendency also for the spin-density polarizability, for which it is reasonable to hold the same interpretative picture. This behaviour is in agreement with studies for various confinements for single- and double-electron dots.9 We remark here that the calculation of spin-density polarizability is not a one-body problem as for the charge-density polarizability, hence it provides a way to put in evidence the effects of the coulombic correlation among the electrons. Moreover, also the charge-density polarizability becomes a many-body problem when the number of components of the dot is increased (around N > 30), since for medium- and large-size dots the confinement has to be anharmonic in order to reproduce real dots. Our method could then be used for calculating both kinds of polarizabilities in larger dots. This would also provide an opportunity for a more realistic comparison with experimental data, which are indeed typically taken on hundreds-electron dots. References 1. 2. 3. 4. 5. 6. 7. 8. 9.

M. Koskinen, M. Manninen and S. M. Reimann, Phys. Rev. Lett. 79, p. 1389 (1997). F. Pederiva, C. J. Umrigar and E. Lipparini, Phys. Rev. B 62, p. 8120 (2000). S. M. Reimann and M. Manninen, Rev. Mod. Phys. 74, p. 1283 (2002). M. Barranco, L. Colletti, E. Lipparini, A. Emperador, M. Pi and L. Serra, Phys. Rev. B 61, p. 8289 (2000). P. A. Maksym and T. Chakraborty, Phys. Rev. Lett. 65, p. 108 (1990). C. Filippi and C. J. Umrigar, Phys. Rev. B 61, p. R16291 (2000). C. Schuller, K. Keller, G. Biese, E. Ulrichs, L. Rolf, C. Steinebach and D. Heitmann, Phys. Rev. Lett. 80, p. 2673 (1998). L. Colletti, F. Pederiva, E. Lipparini and C. J. Umrigar, Phys. Stat. Sol. B 244, p. 2317 (2007). M. Ghosh, R. K. Hazra and S. P. Bhattacharyya, Chem. Phys. Lett. 434, p. 56 (2007).

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PROGRESS IN COUPLED ELECTRON-ION MONTE CARLO SIMULATIONS OF HIGH-PRESSURE HYDROGEN Carlo PIERLEONI INFM-CNR SOFT and Physics Department, University of L’Aquila, L’Aquila, I-67010, Italy ∗ E-mail: [email protected] www.fisica.aquila.infn.it Kris T. DELANEY Materials Research Laboratory, University of California, Santa Barbara, CA 93106-5121, USA E-mail: [email protected] Miguel A. MORALES, David M. CEPERLEY Physics Department, University of Illinois at Urbana-Champaign Urbana, Illinois 61801, USA E-mail: [email protected], [email protected] Markus HOLZMANN LPTMC, Universit´ e Pierre et Marie Curie, 4 Place Jussieu, 75005 Paris, France and LPMMC, CNRS-UJF, BP 166, 38042 Grenoble, France E-mail: [email protected] We report recent progress in the Coupled Electron-Ion Monte Carlo method and its application to high-pressure hydrogen. We describe the particular form of electronic trial wave functions that we have employed in applications for high-pressure hydrogen. We report wave function comparisons for static proton configurations and preliminary results for thermal averages. Keywords: Quantum Monte Carlo; Ab-initio methods; high pressure hydrogen.

1. Introduction Modern ab-initio simulation methods for systems of electrons and nuclei mostly rely on Density Function Theory (DFT) for computing the electronic forces acting on the nuclei, and on Molecular Dynamics (MD) techniques to follow the real-time evolution of the nuclei. Despite recent progress, DFT suffers from well-known limitations.1,2 As a consequence, current ab-initio predictions of metallization transitions at high pressures, or even the prediction of structural phase transitions, are often only qualitative. Hydrogen is an extreme case,3–5 but even in silicon, the diamond/β-

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tin transition pressure and the melting temperature are seriously underestimated.6 An alternative route to the ground-state properties of a many electrons system of is the Quantum Monte Carlo method (QMC).2,7 In QMC, a many-body trial wave function for the electrons is assumed and the electronic properties are computed by Monte Carlo methods. For fermions, QMC is a variational method with respect to the nodes of the trial wave function and a systematic, often unknown, error remains.2,7 Over the years, the level of accuracy of the fixed-node approximation has been improved8–11 such that, in most cases, fixed-node QMC methods have proven to be more accurate than DFT-based methods, on one side, and less computationally demanding than correlated quantum-chemistry strategies (such as coupled cluster method)2 on the other side. Computing ionic forces with QMC to replace the DFT forces in ab-initio MD, poses additional problems whose solution has only very recently been proposed in a consistent way.12,13 In recent years, we have been developing a different strategy, the Coupled Electron-Ion Monte Carlo (CEIMC) method, based entirely on Monte Carlo algorithms, both for solving the electronic problem and for sampling the ionic configuration space.14 The new method relies on the Born–Oppenheimer approximation. A Metropolis Monte Carlo simulation of the ionic degrees of freedom (represented either by classical point particles or by path integrals) at fixed temperature is performed based on the electronic energies computed during independent ground state Quantum Monte Carlo calculations. Application of CEIMC has so far been limited to high pressure hydrogen for several reasons: a) hydrogen is the simplest element of the periodic table, and the easiest to cope with since the absence of the additional separation of energy scales between core and valence electrons as in heavier elements; b) it is an important element since most of the matter in the universe consists of hydrogen; c) its phase diagram at high pressure in the interesting region where the metallization occurs is still largely unknown because present experiments are not able to reach the relevant pressures. We have investigated the very high pressure regime where all molecules are dissociated and the system is a plasma of fully ionized protons and electrons,15 and we have studied the pressure-induced molecular dissociation transition in the liquid phase.16 In both studies the CEIMC results were not in agreement with previous Car-Parrinello Molecular Dynamics (CPMD) calculations.17,18 While we have evidence now that the discrepancy in the fully ionized case is removed by taking our new more accurate trial wave function, in the second study more accurate CEIMC calculations predict a continuous molecular dissociation with increasing pressure at variance with CPMD where a first order molecular dissociation transition was observed by increasing pressure at constant temperature. Recently, using constant volume Born–Oppenheimer Molecular Dynamics rather then constant pressure CPMD, a continuous dissociation transition has been reported from DFT-GGA studies19 In the present paper we will discuss in some details the various trial wave functions we have implemented for hydrogen. In Sec. 2 we briefly review the basic ingredients of the method. Section 3 will be devoted to describing the different trial wave

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functions and some details on their efficient implementation. In Section 4 we will report numerical comparisons among the various trial functions. Finally, in Sec. 5 we collect our conclusions and perspectives. 2. The CEIMC Method In this section we briefly outline the basic ingredients of the CEIMC approach. Further details can be found in several published reviews.14,20,21 CEIMC, in common with the large majority of ab-initio methods, is based on the Born–Oppenheimer separation of “slow” ionic degrees of freedom and “fast” electronic degrees of freedom. In addition, the electrons are considered to be in their ground state which depends on the instantaneous proton positions. Protons, either considered as classical or quantum particles, are assumed to be at thermal equilibrium with a heat bath at fixed temperature T . The system of Np protons and Ne = Np electrons is enclosed in a given volume V which provides a fixed number density n = Np /V , better expressed in terms of the coupling parameter rs = (3/4πn)1/3 . The thermal equilibrium distribution of proton states S, P (S) ∝ e−βEBO (S) is sampled by a Metropolis Monte Carlo calculation.22 Here S is the 3Np -dimensional vector of proton positions and Born–Oppenheimer en the corresponding D EBO (S) E ˆ ˆ ergy defined as EBO (S) = Φ0 (S) H Φ0 (S) where H is the hamiltonian of the system and |Φ0 (S)i its ground state. In order to compute an estimate of EBO (S) we employ both Variation Monte Carlo (VMC) and Reptation Quantum Monte Carlo (RQMC)23 methods within the “fixed node” (for real trial functions) or“fixed phase” (for complex trial functions) approximation.2,14 The bounce algorithm for sampling the electronic paths within RQMC is implemented.14,24 The estimate of EBO (S) for a given trial function computed by QMC is affected by statistical noise which, if ignored, will provide a biased sampling. The size of the bias increases for increasing noise level. A possible solution would be to run very long QMC calculations in order to get a negligibly small noise level, and thus a negligible bias. However, for each protonic configuration S, the noise level decreases as the square root of the number of independent estimates of EBO (S). This means that in order to decrease the noise level by one order of magnitude we should generate 100 times more uncorrelated samples, an unfavorable scaling especially given that the process must be repeated for any attempted move of the ions. The less obvious, but far more efficient, solution is to generalize the Metropolis algorithm to noisy energies. One such algorithm is the Penalty Method.14,25 The idea is to require the detailed balance to hold on average (over the noise distribution) and not for a single energy calculation. Within the Penalty Method the acceptance probability of a single protonic move S → S 0 depends, not only on the energy difference between the two states β[EBO (S 0 ) − EBO (S)], but also on the noise of the energy difference (βσ)2 . Here β is the inverse physical temperature of the protons. Since (βσ)2 > 0 the presence of the noise always causes extra rejection of attempted moves with respect

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to the noiseless case. It is clear that the method is successful if one can allow for a large and cheap noise level still keeping a nonvanishing acceptance probability (≤ 0.1). A rule of thumb for maximum efficiency is to have (βσ) ∼ 1.25 An efficient energy difference method is exploited to compute the energy difference and its noise.14,20 A well known problem for QMC energies, in particular for metals, is caused by finite size effects, mainly coming from the discrete nature of the reciprocal space of finite and, generally small, systems. For a degenerate Fermi liquid, finite-size shell effects are much reduced if twist averaged boundary conditions (TABC) are used. 26 ˆ the TABC is defined as For a given property A, Z π dθ~ ˆ = ˆ ~i hAi hΨθ~ |A|Ψ (1) θ d −π (2π) where ~ θ is a 3D-vector specifying the undetermined phase that the N -body wave ~ function Ψθ~ (~r1 + L~n, ~r2 , . . .) = eiθ Ψθ~(~r1 , ~r2 , . . .) picks up when a particle wraps around the boundary of the simulation box. TABC is particularly important in computing properties that are sensitive to the single particle energies, such as the kinetic energy and the magnetic susceptibility. By reducing shell effects, accurate estimates of the thermodynamic limit for these properties can be obtained already with a limited number of electrons. In CEIMC, we can take advantage of twist averaging to reduce the noise in the energy difference for the acceptance test of the penalty method. Different strategies can be used to implement the TABC. One possibility is to use a fixed 3D grid in the twist angle space, at each grid point run independent QMC calculations and then average the resulting properties. However, the optimal noise level in CEIMC is βσ ∼ 1 and a limited number of twists are able to satisfy this requirement at high temperature. At the same time, too coarse a grid introduces a systematic effect on the energy of the system. To illustrate this fact we report in panel a) of Fig. 1 for a single pair of configurations of 16 protons at rs = 1 and T = 3000K, the energy difference as obtained with VMC and the metallic wave function (see Sec. 3). The energy difference is indeed the key quantity in CEIMC since it guides the Metropolis sampling of the protonic degrees of freedom. As can be seen, the energy difference computed over a fixed grid has an oscillating behavior with the number of twists, which implies that convergence requires a large numbers of twists. Running that large number of twists will be very time consuming and will provide a noise level much smaller than the optimal value. To solve this problem of efficiency, we can think of the twist angle as an additional random variable to be sampled during the protonic MC simulation. To this aim, we still use a fixed grid in the twistangle space, but at each protonic step we sample a value of the twist angle inside the Wigner-Seitz cell around each grid point. In panel a) of Fig. 1, we also report the value of the energy difference obtained sampling the twists for the same pair of proton configurations. We see that the energy difference converge much more rapidly with the number of the twists, as expected. The conclusive test on the accuracy of

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6x6x6 FT 4x4x4 ST 2

fixed twists sampled twists

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3.2

gpp(r)

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3.1

1

3 0.5 2.9

b)

a) 2.8

1

10

100 #twists

1

2

3

4

0

r/a0

Fig. 1. a) Reduced energy difference for a pair of protonic configurations of a 16 protons system at rs = 1 sampled during a run at T = 3000K versus the number of twist angles. The energy difference is obtained with the metallic trial function and at the VMC level. Closed circles corresponds to fixed twists on a Monkhorst-Pack grid with 2x2x2, 4x4x4, 6x6x6, 8x8x8 and 10x10x10 (with inversion symmetry), while squares corresponds to twists sampled as explained in the text. b) proton-proton pair correlation function for a system of 54 protons at rs = 1 and T = 1500K with the metallic trial function at the VMC level. Closed circles are data for fixed twists on a 6x6x6 grid while squares are for 32 sampled twists.

the twist sampling is, however, provided only by comparing equilibrium CEIMC calculations with fixed and sampled twists. This is illustrated in panel b) of Fig. 1 for 54 protons at rs = 1 and T = 1500K, comparing the the proton-proton pair correlation functions obtained by using 108 fixed twists and 32 sampled twists at the VMC level and with the metallic wave function (see Sec. 3). 3. Trial Wave Functions for Hydrogen In the first implementation of CEIMC,20,21 our goal was to simulate the insulating phases of molecular hydrogen and, as such, a trial function consisting of a few optimizable guassian molecular orbitals centered on each molecule was used. Optimization of the variational parameters, in number proportional to the number of electron in the system, needed to be performed at each ionic configuration and was a major bottleneck for the efficiency of the method. Subsequently, we have developed trial functions with a very limited number of variational parameters (even zero when possible) and therefore reduce the complexity of the optimization step in the electronic calculation (or to reduce to a linear optimization in the case of DFT orbitals). In Ref. 10 we have shown, as the Feynman–Kac formula suggests, a procedure to

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iteratively improve any initial trial function. If a Hartree–Fock (HF) determinant is assumed as an initial ansatz , the first iteration generates a bosonic (symmetric) twobody correlation function (Jastrow) while the next iteration naturally provides the backflow transformation of the orbitals and a three-body bosonic correlation term. Unfortunately, this is a formal theory which cannot provide, in general, analytical expressions for the various terms. Nonetheless the general structure is illuminating in searching for improvements. 3.1. The metallic wave function At a very high pressure, beyond metallization and molecular dissociation, the electron liquid is a good Fermi liquid and correlation effects, with protons and among electrons, can be treated as perturbations. In this case it is natural to assume a determinant of free electron states (plane waves) as an initial ansatz . An accurate and paA rameter free Jastrow factor can be obtained within the RPA uRP (r) (i, j = e, p).27 ij This simple form satisfies the correct cusp conditions at short particle separations and the right plasmon behavior (screening) at large distances. It was shown28 to provide good energies for hydrogen even at intermediate densities if supplemented 2 2 A by Gaussian functions u ˜ij (r) = uRP (r)−αij e−r /wij , with the variational parameij ters αij , wij . The additional term preserves the short- and long-distance behavior of the RPA function and corrects for possible inaccuracies at intermediate distances. However, they introduce four variational parameters, namely αee , wee , αep , wep . As stated above, the next iteration suggests the backflow transformation of the orbitals and a three-body correlation factor. This is a crucial step for an inhomogeneous electron system, since the nodal surfaces of the trial wave function will become explicitely dependent on the proton positions and will provide a more accurate energy even at the RQMC level. Similar to the case of the homogeneous electron gas, 9 the backflow and three-body functions were at first parametrized as Gaussians.20 This trial function has a total of 10 free parameters to be variationally optimized and has been used in a first CEIMC study of the melting transition of the proton crystal in hydrogen at rs = 1.20 Next, we were able to derive approximate analytical expressions for the backflow and the three-body functions, as well as for the two-body correlation factor, in the Bohm–Pines collective coordinates approach.10 This form is particularly suitable for the CEIMC because it is parameter-free. At the same time, it provides comparable accuracy to the numerically optimized wave function, both in the crystal configuration and for disordered protons (see Sec. 3.2). Explicit forms of the various terms can be found in the appendix of Ref. 10. With this kind of wave function we have investigated the melting at three densities (rs = 0.8, 1.0, 1.2) including quantum effects for protons.14,15 3.2. Band-structure-based wave functions (IPP/LDA) The metallic wave function is expected to provide an accurate description of the electronic ground state at high density, well beyond molecular dissociation and

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metallization. On the other hand we expect it to be a poor representation of the true ground state at lower densities where molecules appear and plane-wave singleelectron orbitals (although in terms of backflow coordinates) are certainly not a good representation. Natoli et al.29,30 have previously used a Slater determinant of Kohn–Sham self consistent orbitals to study the solid phases of atomic hydrogen at rs = 1.31 and T = 0, and of molecular hydrogen at lower densities. They have found a typical energy gain of 0.5eV/electron by replacing the plane-wave with the self consistent orbitals in the Slater determinant. Here we have implemented similar ideas. The single-particle orbitals, {φn }, that comprise the Slater determinant, are computed on-the-fly during the CEIMC calculation as the eigenstates of some singleparticle Hamiltonian, ˆ n (r) = − 1 ∇2 + Veff (r; S) φn (r) = εn φn (r) , (2) hφ 2 where the N/2 orbitals with the lowest eigenvalue, εn , are selected to fill the deterˆ describes electron-nuclear interactions minants. The single-particle Hamiltonian h and approximates electron-electron interactions through an effective potential. To solve the eigenvalue problem we use an iterative, conjugate-gradients bandby-band minimization scheme.31 The method employs the variational principle to minimize residuals, and the Gram-Schmidt scheme to preserve orthogonalization of the eigenstates. In our studies, we assume either Veff = Ve−n , the bare electron-nuclear interaction (IPP independent particle potential), or Veff = VLDA , the Kohn–Sham effective potential within the local density approximation (LDA) with the Perdew-Zunger32 parameterization of Ceperley-Alder33 electron-gas data. In both cases, the wave functions are eigenstates of a Hamiltonian which contains a bare Coulomb interaction between electrons and protons. The singularity in the potential results in a derivative cusp in the orbitals −∂ln[φ (r)]/∂r = 1 for r = RI . This cusp is important for obtaining good energies or short projection times for QMC algorithms. Representing this cusp on a plane wave basis is challenging due to the slow algebraic decay of k −4 . For this reason, we implement a cusp-removal method by dividing the orbitals by a function that satisfies the cusp condition exactly (in this case, the RPA ep Jastrow function discussed earlier), before reverse Fourier transforming to retrieve the plane wave coefficients of the orbitals used to build the Slater determinant. The proper electron-proton cusp is later reintroduced using the same RPA Jastrow function. This procedure greatly enhances the convergence of the SlaterJastrow wave function with respect to the size of the plane-wave basis set used to represent the orbitals 3.3. Backflow transformations of IPP/LDA orbitals Further complexity may be introduced into these trial functions through the use of a backflow transformation which introduces correlations into the fermionic part of a Slater-Jastrow wave function, with the advantage that modifications of the

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nodal surface are possible. As previously discussed, analytic forms for the backflow function have been derived using the Bohm–Pines collective coordinates approach. 10 While this form is strictly valid for plane-wave states in a determinant, we could think of applying the Feynman–Kac iteration to generate a backflow transformation even for IPP or LDA orbitals. The specific form for the transformation is unknown, but as a first ansatz we can use the same expressions we have developed in the metallic case. The obtained trial functions will be denoted IPPBF and LDABF respectively. As we will show in the next section, this procedure is found always to improve the total energy and its variance, providing therefore a better representation of the ground state of the system.

4. Comparisons of Wave Functions 4.1. Fixed protons configurations In this section we consider fixed proton configurations and we compare the quality of the various wave functions at two densities, corresponding to rs = 1.0 and rs = 1.4. Results for hydrogen at rs = 1.31 in the bcc structure obtained from various improvements of the metallic wave function are reported in Table III of Ref. 10 where they are also compared to the results obtained with self-consistent Kohn–Sham orbitals.29 There we have shown that the quality of the analytical form of the metallic wave function is superior to its numerically optimized version and comparable to that of the LDA orbitals for hydrogen in the bcc structure and for various system sizes. Our present implementation of LDA orbitals provides results in agreement with previous estimates.34 In Table 1 we report QMC energies for hydrogen in several crystal structures and for various system sizes. A complete study of the size dependence and the relative stability of those structure is not our concern here and will be reported elsewhere. 34 From Table 1 we observe that LDA always provides a small or neglible improvement over IPP, while IPP is significantly cheaper through the lack of the self-consistent requirement. Comparing various structures and system sizes, we observe that the best wave function depends on the structure: for bcc, fcc structures and the diamond structure with N=8, the metallic wave function is superior to the others. The opposite is true for the diamond structure with N=64, where IPP and LDA provide lower energies at all densities. We also observe that the ordering of wave functions does not appear to depend on density, at least in the limited range investigated. Note that rs = 1.31 corresponds to the density predicted by ground state QMC calculations28 for the molecular dissociation to occur. Another important test is the effect of backflow (BF) on the band orbitals. We see that both at the variational and at the reptation level the energy is slightly improved and at the same time the variational variance is halved by the backflow, which means a net improvement of the variational function and the need of a shorter projection in imaginary time to reach the ground state. The high level of accuracy observed for the metallic

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rs = 1.0

rs = 1.4

WFS

Ev

σv2

Er

σr2

Ev

σv2

Er

σr2

Met

-0.36931(1)

0.0279(2)

-0.3721(1)

0.0182(7)

-0.5203(2)

0.036(2)

-0.5224(1)

0.01008(4)

IPP

-0.3681(3)

0.0765(3)

-0.5139(1)

0.0887(2)

-0.5209(2)

0.0284(1)

LDA

-0.3681(2)

0.0765(2)

-0.5145(1)

0.0869(3)

-0.5210(2)

0.0286(1)

IPP+BF

-0.3705(1)

0.0359(1)

-0.5228(1)

0.01413(7)

-0.3705(1)

0.0357(1)

-0.3792(1)

0.01543(4)

-0.5272(1)

0.00872(3)

-0.36805(2)

0.04636(4)

Met fcc

IPP

-0.3756(2)

0.0756(2)

-0.5210(1)

0.0835(3)

-0.5256(1)

0.0276(1)

(Np = 32)

LDA

-0.3757(2)

0.0753(2)

-0.5212(1)

0.0828(3)

-0.5259(1)

0.02724(9)

IPP+BF

-0.3779(1)

0.03550(9)

-0.5280(1)

0.01352(5)

LDA+BF

-0.3779(1)

0.0351(1) -0.5168(1)

0.01656(8)

-0.5321(5)

0.0339(3)

DIAM (Np = 64)

Met

-0.3477(2)

0.0268(1)

IPP

-0.3621(2)

0.0830(4)

-0.5189(2)

0.1027(8)

LDA

-0.3613(2)

0.0823(3)

-0.5323(1)

0.0331(2)

IPP+BF

-0.3637(1)

0.0404(1)

-0.5346(1)

0.01740(7)

LDA+BF

-0.3635(1)

0.0406(1)

-0.41060(4)

0.02136(4)

-0.41368(6)

0.01032(2)

IPP

-0.40198(8)

0.0863(1)

-0.4094(1)

0.04342(8)

LDA

-0.40206(8)

0.0865(1)

-0.4098(1)

0.04356(6)

IPP+BF

-0.40632(6)

0.04958(8)

-0.41070(6)

0.02382(4)

LDA+BF

-0.40638(6)

0.04958(8)

-0.4107(1)

0.02382(6)

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Table 1. The energy and variance of hydrogen in various structures with different trial functions. All results are obtained averaging over a 6x6x6 fixed grid of twist angles. Ev and σv2 represent, respectively, energy and variance at the variational level while E r and σr2 are the energy and the mixed estimator for the variance obtained with RQMC. Units are Hartree/atom.

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0.002

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-0.37 0.0005 Metallic VMC Metallic RQMC LDABF VMC LDABF RQMC

-0.38 0

5

10 # conf

15

Metallic LDABF 0

5

10 # conf

15

0

Fig. 2. Total energy (left panel) and quality parameter (right panel) for a number of static proton configurations as obtained with the metallic and the LDABF trial functions at r s = 1. TABC with a 6x6x6 fixed grid in the twist space is performed. Energies are in h/atom.

wave function induced us to perform a detailed study of liquid hydrogen at finite temperature.15 Next, we consider how the various wave functions perform on disordered protonic configurations representative of atomic hydrogen in the liquid state. As before, all results reported here are averaged over a 6x6x6 fixed grid in the twist space. At rs = 1 we compare the Metallic wave function with the LDABF wave function, while at lower density we report data for the IPP, IPPBF and the LDABF wave functions. Data for configurations at rs = 1 are presented in Figure 2. We display on the left panel VMC and RQMC energies for 18 protonic configurations obtained with the metallic and the LDABF wave functions. Configuration 0 is a 32 protons warm crystal near melting (fcc), configuration 1 is the perfect bcc crystal with 54 protons, configurations 2 to 12 are statistically independent configurations of 54 protons obtained during a CEIMC run at T=2000K performed with the LDABF trial function, while the remaining 5 configurations have been obtained during a CEIMC run at the same temperature performed with the metallic trial function. The panel on the right reports the values of the quality parameter a . Several interesting facts can be inferred from this figure. With the noticeable exception of the perfect bcc a The

quality parameter of a trial function is defined as the negative logarithm of the overlap of the trial state onto its fully projected state. It is easy to prove that it reduces to the integral over the positive imaginary time axis of the difference between the energy and its extrapolation at infinite time. The smaller the quality parameter the better the trial function is.

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crystal, energies from LDABF wave function are always lower than energies from the metallic wave function. In particular, the fully converged RQMC energies from the metallic wave functions are above the VMC energies from LDABF. This implies that changing the form of the nodes provides more energy than fully projecting the initial state. Why the excellent quality of the metallic wave function observed in perfect crystals is deteriorated by disordered remains unclear to us, but as a matter of fact it appears that LDA nodes supplemented by e-e backflow perform much better both in the liquid state and in the crystal state with thermal fluctuations. Another interesting observation concerns the dispersion of the energies over a set of configurations. Let us consider the first 11 liquid configurations (from 2 to 13) generated during a run with the LDABF trial function. Considering the RQMC energies, the dispersion is 1.380(3) mH/atom with the metallic wave function but only 0.730(2) mH/atom with the LDABF wave function (see horizontal dashed lines in the left panel of Fig. 2), that is the BO surface with LDABF is smoother than the other and the liquid will be less structured (see the next section). Finally, it is interesting to compare the dispersion of the VMC and the RQMC energies for a given trial function and a given set of configurations. Always for the first 11 liquid configurations and for the LDABF wave function we have 0.820(1) mH/atom at the VMC level versus 0.730(2) mH/atom at RQMC level. This implies that projecting the trial wave function will only provide a tiny difference in the roughness of the BO energy surface (corresponding to a temperature effect of ∼ 0.09mH/atom = 30K). As for the quality parameter, we similarly observe that, with the exception of the bcc crystal, the metallic wave function has larger values, which means that it is less accurate than the LDABF wave function. Note also, how the quality of the LDABF wave function is uniform (at fixed number of particles) through the perfect crystal and the disordered configurations, no matter how these configurations have been generated. This is an important requirement to accurately predict phase transitions. On the other hand, the quality of the metallic wave function on the 5 liquid configurations generated with this wave function is higher than on the remaining 10 liquid configurations generated in a LDABF run. A good trial function should have a uniform quality throughout the entire proton configurational space in order to provide an unbiased sampling. A similar analysis has been performed at rs = 1.4 considering 5 uncorrelated liquid configurations generated during a CEIMC run at T = 2000K with the LDABF trial function. Results are displayed in Fig. 3. Since the metallic wave function is certainly not accurate at this density, we compare the IPP and LDABF wave functions only. Again, the quality of LDABF is superior to the quality of the other wave function because it has a lower energy (its VMC energy is very close to the RQMC-IPP energy) and a smaller and more uniform quality parameter. As for the dispersion of the energy at the RQMC level we obtain 4.762(2) mH/atom for LDABF and 5.796(2) mH/atom for IPP, suggesting that the liquid structure at a given temperature (and in particular the molecular fraction) could depend on the trial functions. Finally the VMC and RQMC energy dispersions for LDABF are 4.862(1)mH/atom and 4.762(2)mH/atom respectively, suggesting, as

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0.004 -0.5 0.0035

IPP VMC IPP RQMC LDABF VMC LDABF RQMC

-0.505

Etot (h/atom)

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0.003

-0.51 0.0025 -0.515 IPP LDABF

-0.52

0.002

0.0015

-0.525

0.001

-0.53 0

1

2 # conf

3

4

0

1

2 # conf

3

4

Fig. 3. Total energy (left panel) and quality parameter (right panel) for a number of static proton configurations as obtained with the metallic and the LDABF trial functions at r s = 1.4. TABC with a 6x6x6 fixed grid in the twist space is performed. Energies are in h/atom.

before, that projecting the trial wave function will only slightly change the protonic structure, the larger effect being in changing the nodal structure. 4.2. Liquid-state simulations After the validation of LDABF trial function of the previous section, we report here results for the liquid structure of hydrogen at the same densities. We have simulated systems of 54 protons. The TABC is performed here using twist sampling around the nodes of a 4x4x4 grid in twist space at each protonic step. In the left panel of Fig. 4 we report a comparison of proton-proton pair correlation functions gpp (r) at rs = 1 and T = 1500K as obtained from the metallic and LDABF trial functions at the VMC level. As expected from the results of the previous section, we observe considerably more structure with the metallic trial function than with the LDABF one, which indeed would correspond to having an effective lower temperature. On the same figure we report results from a CPMD simulation17 performed within the LDA approximation. The agreement between CPMD data and our present CEIMC data from LDABF trial function is striking and somehow unexpected. Indeed our representation of the electronic ground state is much more accurate than the simpler LDA one. Also the finite size effect in the CPMD calculation was addressed only partially by using only closed shells systems at the Γ point. Nonetheless the observed agreement testify that the structure of the proton liquid is not very sensitive to

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1.8 Metallic LDABF CPMD

2

IPP LDABF

1.6 1.4 1.2 1 0.8

gpp(r)

1.5 gpp(r)

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1 0.6 0.4

0.5

0.2 0

1

2

3 r/a0

4

1

2

3 r/a0

4

5

0

Fig. 4. Left panel: rs = 1, T = 1500K, Np = 54. Proton-proton pair correlation functions as obtained with LDABF and metallic wave functions. Results are obtained with TABC by using twist sampling around a 4x4x4 grid. CPMD data from ref.17 are also represented by a thick dashed line. Right panel: rs = 1.4, T = 2000K, Np = 54. Proton-proton pair correlation functions as obtained with IPP and LDABF wave functions. Results are obtained with TABC by using a 6x6x6 fixed grid (IPP) and by twist sampling around a 4x4x4 grid (LDABF).

details of the ground state representation. Finally, in the right panel of Figure 4 we report preliminary data for gpp (r) of 54 protons at rs = 1.4 and T = 2000K. We compare IPP and LDABF trial functions at the VMC level. The statistical noise is still large but it seems that the overall behavior does not depend too much on the kind of trial functions, although small details could still be different. Note, however, that the liquid has little structure. More investigations of the influence of the trial function on the liquid structure is certainly needed, in particular, in the molecular dissociation region.

5. Conclusion We have reported important progress in CEIMC, an efficient and accurate method to perform ab-initio simulations of condensed system with QMC energies. We have shown how the method performs in the case of hydrogen at high pressure, the simplest, but yet not understood, system.The new method allows us to cover a range of temperatures inaccessible to previous QMC methods for hydrogen, a range where most of the interesting physics of hydrogen occurs, including the melting of the molecular and proton crystals, the molecular dissociation both in the liquid and in the crystal and the metallization of the system.

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A key ingredient in CEIMC is the trial function used to represent the electronic ground state. Even when a projection technique such as Reptation QMC is exploited to improve the bosonic part of the trial many-body wave function, its fermionic part, that is its nodal surface, is still playing a very crucial role in determining the electronic energies and therefore the overall thermal behavior of the system. In the present paper, we have reported a detailed investigation of these effects for hydrogen by comparing a number of different trial wave functions at two densities. We have shown as a fully analytical trial wave function, that is optimal in terms of computational efficiency in CEIMC, and which has been previously demonstrated to provide excellent accuracy for crystalline states, degrades as soon as some disorder is introduced in the protonic configurations. This result has been established by comparing with results for new trial functions obtained from a Slater determinant of IPP/LDA orbitals together with a two-body Jastrow correlation factor. A further backflow transformation of these orbitals has been introduced and characterized. The new trial functions provide lower energies and more uniform overlap over a number of fixed representative configurations, which we use as an indication of the overall quality of the trial function. The most striking result on disordered configurations is that the LDABF energies at the VMC level are lower than the fully projected energies from the metallic trial function. This indicates that the improvement of performance comes mainly from the different nodal surfaces, while the bosonic part is responsible only for smaller improvements. The failure of the metallic wave function is most probably due to the presence of some degeneracy of its orbital structure around the Fermi surface which is removed by solving the instantaneous band structure. On the other hand, the use of complex wave functions and twist averaged boundary conditions in connection with the metallic trial function was expected to remove most of these degeneracies. A better understanding of this failure is desirable and deserves more investigation. The difference in energies for different trial functions, or more precisely the dispersions of the energies from different wave functions, translates in a overall temperature factor at thermal equilibrium. The metallic trial function provides a dispersion which is roughly twice that of the corresponding dispersion from the LDABF trial function. Therefore the metallic gpp (r) at temperature T should correspond to the LDABF gpp (r) at ∼ T /2. This is indeed observed and the new gpp (r)’s from LDABF are in fair agreement with predictions of Car-Parrinello MD.17 This agreement remains somehow surprising since, beyond the different methods of sampling protonic configurational space, the electronic description in the two methods is quite different. We use LDA orbitals with a backflow transformation and a two body RPA Jastrow while in CPMD, only LDA orbitals are employed. Adding the backflow and the Jastrow we obtain a fair gain of energy and moreover we can improve the bosonic part of the trial function by projecting in imaginary time. Further we strongly reduce the finite size effects by averaging over the undetermined phase of the wave function, while CPMD calculations are performed at the Γ point only for closed shell systems (Np = 54 and 162). However the final agreement between the

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two methods indicated that the effects of these improvements on the energy difference is only minor. On the other hand, it is well known in simple liquids that g(r) is not very sensitive to changes of the interaction potential and this might explain the observed agreement. At lower densities, employing IPP orbitals and RPA Jastrow, we have recently found16 a continuous molecular dissociation with density, at variance with CPMD which has predicted a first order molecular dissociation transition.18 The reliability of IPP trial function was only tested on crystal structures and should be further investigated for disordered configurations along the lines shown here. This study is in progress. A recent BOMD study19 within DFT/GGA has reported a continuos molecular dissociation in agreement with our findings. This agreement suggests that improving the trial functions from IPP to LDABF might change the details of the results but not the overall picture. This confirms that our present method can be most useful in condition where new interesting physics is happening, such as near a liquid-liquid phase transitions or a metallization transition. Acknowledgments C.P. acknowledges financial support from MIUR-PRIN2005, D.M.C. and K.T.D. acknowledge support from DOE grant DE-FG52-06NA26170, M.A.M. acknowledges support from a DOE/NNSA-SSS graduate fellowship and M.H. acknowledges support from ACI “Desorde et Interactions Coulombiennes” of French Ministry of Research and from CNISM, Italy. D.M.C. and M.H. acknowledge support from a UIUC-CNRS exchange program. Computer time was provided by NCSA and CINECA. References 1. R. M. Martin, Electronic Structure. Basic Theory and Practical Methods (Cambridge University Press, Cambridge, 2004). 2. M. W. C. Foulkes, L. Mitas, R. J. Needs and G. Rajagopal, Rev. Mod. Phys. 73, p. 33 (2001). 3. E. G. Maksimov and Y. I. Silov, Physics-Uspekhi 42, p. 1121 (1999). 4. M. Stadele and R. Martin, Phys. Rev. Letts. 84, 6070 (2000). 5. K. A. Johnson and N. W. Ashcroft, Nature 403, p. 632 (2000). 6. D. Alfé, M. Gillan, M. Towler and R. Needs, Phys. Rev. B 70, p. 161101 (2004). 7. B. L. Hammond, W. A. Lester and P. J. Reynolds, Monte Carlo methods in Ab Initio Quantum Chemistry (World Scientific, Singapore, 1994). 8. R. M. Panoff and J. Carlson, Phys. Rev. Letts. 62, p. 1130 (1989). 9. Y. Kwon, D. Ceperley and R. Martin, Phys. Rev. B 50, 1684 (1994). 10. M. Holzmann, D. M. Ceperley, C. Pierleoni and K. Esler, Phys. Rev. E 68, 046707 (2003). 11. P. L. Rios, A. Ma, N. D. Drummond, M. Towler and R. J. Needs, Phys. Rev. E 74, 066701 (2006). 12. J. C. Grossman and L. Mitas, Phys. Rev. Letts. 94, 056403 (2005). 13. S. Sorella and C. Attaccalite, Ab initio molecular dynamics for high-pressure liquid hydrogen, unpublished, (2007).

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14. C. Pierleoni and D. M. Ceperley, The Coupled Electron-Ion Monte Carlo Method, in Computer Simulations in Condensed Matter: from Materials to Chemical Biology, eds. M. Ferrario, G. Ciccotti and K. Binder, Lecture Notes in Physics, Vol. 703 (SpringerVerlag, 2005), pp. 641–683. 15. C. Pierleoni, D. Ceperley and M. Holzmann, Phys. Rev. Letts. 93, 146402 (2004). 16. K. T. Delaney, C. Pierleoni and D. M. Ceperley, Phys. Rev. Letts. 97, 235702 (2006). 17. J. Kohanoff and J. P. Hansen, Phys. Rev. E 54, p. 768 (1996). 18. S. Scandolo, Proc. Natl. Acad. Sci. U.S.A. 100, p. 3051 (2003). 19. J. Vorberger, I. Tamblyng, B. Militzer and S. A. Bonev, Phys. Rev. B 75, p. 024206 (2007). 20. D. M. Ceperley, M. Dewing and C. Pierleoni, The Coupled Electron-Ion Monte Carlo Simulation Method, in Bridging Times Scales: Molecular Simulations for the Next Decade, eds. P. Nielaba, M. Mareschal and G. Ciccotti, Lecture Notes in Physics, Vol. 605 (Springer-Verlag, 2002), pp. 473–499. 21. M. Dewing and D. M. Ceperley, The Coupled Electron-Ion Monte Carlo Simulation Method, in Recent advances in Quantum Monte Carlo Methods II , eds. W. A. Lester, S. M. Rothstein and S. Tanaka (World Scientific, 2002). 22. D. Frenkel and B. Smit, Understanding Molecular Simulations: From Algorithms to Applications, 2nd edn. (Academic Press, San Diego, 2002). 23. S. Baroni and S. Moroni, Reptation quantum Monte Carlo, in Quantum Monte Carlo Methods in Physics and Chemistry, eds. M. Nightingale and C. Umrigar (Kluwer, 1999), p. 313. 24. C. Pierleoni and D. Ceperley, ChemPhysChem 6, 1872 (2005). 25. D. M. Ceperley and M. Dewing, J. Chem. Phys. 110, p. 9812 (1999). 26. C. Lin, F. H. Zong and D. M. Ceperley, Phys. Rev. E 64, 016702[1 (2001). 27. T. Gaskell, Proc. Phys. Soc. London 77, p. 1182 (1961). 28. D. M. Ceperley and B. J. Alder, Phys. Rev. B 36, p. 2092 (1987). 29. V. D. Natoli, R. M. Martin and D. M. Ceperley, Phys. Rev. Letts. 70, p. 1952 (1993). 30. V. D. Natoli, R. M. Martin and D. M. Ceperley, Phys. Rev. Letts. 74, p. 1872 (1995). 31. M. Payne, M. Teter, D. Allan, T. Arias and J. Joannopoulos, Rev. Mod. Phys. 64, p. 1045 (1992). 32. J. Perdew and A. Zunger, Phys. Rev. B 23, p. 5048 (1981). 33. D. Ceperley and B. Alder, Phys. Rev. Lett. 45, p. 566 (1980). 34. K. T. Delaney and et al., to be published .

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QUANTUM PHASE TRANSITIONS ON PERCOLATING LATTICES ´ A. HOYOS THOMAS VOJTA and JOSE Department of Physics, University of Missouri-Rolla, Rolla, Missouri 65409, USA E-mail: [email protected] When a quantum many-particle system exists on a randomly diluted lattice, its intrinsic thermal and quantum fluctuations coexist with geometric fluctuations due to percolation. In this paper, we explore how the interplay of these fluctuations influences the phase transition at the percolation threshold. While it is well known that thermal fluctuations generically destroy long-range order on the critical percolation cluster, the effects of quantum fluctuations are more subtle. In diluted quantum magnets with and without dissipation, this leads to novel universality classes for the zero-temperature percolation quantum phase transition. Observables involving dynamical correlations display nonclassical scaling behavior that can nonetheless be determined exactly in two dimensions. Keywords: Disorder; percolation; quantum magnet; quantum phase transition.

1. Introduction In disordered quantum many-particle systems, random fluctuations due to impurities and defects coexist with quantum fluctuations and thermal fluctuations. Close to phase transitions , the interplay between these different types of fluctuations can cause many unconventional phenomena such as quantum Griffiths effects,1,2 infinite-randomness critical points3,4 and the destruction of the phase transition by smearing5 (for a recent review see, e.g., Ref. 6). A particular interesting case of this scenario are randomly diluted magnets . Site or bond dilution defines a percolation problem7 for the lattice which can undergo a geometric phase transition between a disconnected and a percolating phase. Here, we discuss how the interplay between the quantum fluctuations of the spins and the geometric fluctuations of the lattice changes the phase diagram and the nature of the magnetic percolation transition. Our paper is organized as follows: In Sec. 2, we collect the basic results of classical percolation theory 7 to the extent necessary for the following sections. Section 3 is devoted to the behavior of classical magnets on percolating lattices. These are older results summarized here mainly for comparison with the quantum case. Section 4 is the main part of the paper. Here, we discuss three different examples of percolation quantum phase transitions (QPTs) of diluted quantum magnets. We conclude in Sec. 5.

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Fig. 1. Site-diluted square lattice at impurity concentrations below (p = 0.3), at (p = p c ≈ 0.4073), and above (p = 0.6) the percolation threshold.

2. Geometric Percolation Classical percolation theory7 deals with the geometric properties of randomly diluted lattices. The central question is whether the diluted lattice contains a large cluster that spans the entire sample or whether it is decomposed into small disconnected pieces. For definiteness, consider a d-dimensional hypercubic lattice with bonds between the nearest neighbor sites in which a fraction p of all sites (site percolation) or bonds (bond percolation) is removed at random. Depending on p, the lattice can be in one of two “phases”, separated by a sharp percolation threshold at p = pc . If p < pc (the percolating phase), there is a single large cluster that spans the entire sample (as well as some smaller clusters). In the thermodynamic limit, this cluster, the so-called infinite cluster, becomes infinitely large and contains a nonzero fraction P∞ of all sites. In contrast, for p > pc the lattice is decomposed into small disconnected finite-size clusters only (see Fig. 1). Right at p = pc , there are clusters on all length scales, and their structure is fractal. The behavior of the diluted lattice close to pc is very similar to the critical behavior near a continuous (2nd order) phase transition with the geometric fluctuations due to dilution playing the role of the usual thermal or quantum fluctuations. This implies that observables are governed by power-law scaling relations. A central quantity is the cluster size distribution ns (p). It measures the number of connected clusters with s sites (per lattice site). Close to pc , it takes the scaling form ns (p) = s−τc f [(p − pc ) sσc ] ,

(1)

where σc and τc are critical exponents . In two dimensions, they are known exactly, σc = 36/91 and τc = 187/91; and in three dimensions they are well known numerically, σc ≈ 0.45 and τc ≈ 2.18. f (x) is a scaling function which behaves as f (x) ∼ exp(−B1 x1/σc ) for x > 0, f (x) = const for x = 0, and f (x) ∼ exp[−(B2 x1/σc )1−1/d ] for x < 0. The critical behavior of all other geometric properties can be expressed in terms of the exponents σc and τc . In particular, in the percolating phase, the fraction of sites in the infinite cluster behaves as P∞ ∼ (pc − p)βc with the exponent βc given by βc = (τc − 2)/σc . When approaching the percolation threshold, the typical linear size of the finite-size clusters, the

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Fig. 2. Schematic phase diagrams for classical and quantum magnets on diluted lattices. In the classical case, magnetic order is destroyed by increasing the temperature T , in the quantum case by increasing the quantum fluctuations, e.g., the transverse field hx in a quantum Ising magnet.

connectedness length, diverges as ξc ∼ |p − pc |−νc with νc = (τc − 1)/(dσc ). Finally, the fractal dimension Df of the infinite cluster at the percolation threshold can be expressed as Df = d/(τc − 1). 3. Diluted Classical Magnets In this section we briefly summarize the behavior of a classical Ising or Heisenberg magnet on a randomly diluted lattice. Consider the Hamiltonian X i j Si Sj , (2) H = −J hi,ji

where Si is a classical Ising or Heisenberg spin at site i, and J > 0 is the exchange interaction between nearest neighbors. The dilution is implemented via quenched random variables i taking the values 0 and 1 with probabilities p and 1 − p, respectively. We first discuss the temperature-dilution phase diagram. In the absence of dilution, the model orders ferromagnetically below a critical temperature Tc (0) (provided d ≥ 2 for Ising and d ≥ 3 for Heisenberg spins). Upon dilution, magnetic order is weakened, and Tc decreases. An important question is whether the magnetic phase is completely destroyed before the dilution reaches pc , right at pc , or whether long-range order survives even on the critical percolation cluster at p c , corresponding to phase diagrams (a), (b), and (c) in Fig. 2, respectively (long-range order cannot survive for p > pc because the system consists of small disconnected clusters). Phase diagram (a) can be excluded because the infinite percolation cluster is a massive d-dimensional object for any p < pc . Since the critical percolation cluster at p = pc has a fractal dimension Df > 1, one might be tempted to conclude that it supports magnetic long-range order, at least in the Ising case (implying a phase diagram of type (c)). However, this is incorrect: At the percolation threshold, thermal fluctuations immediately destroy the magnetic order.8–10 This can be understood by considering the “red sites” of the critical percolation cluster, i.e., sites that divide the cluster into two otherwise disconnected pieces (see Fig. 3). The orientation of the spins on these two pieces can be flipped with respect to each other with a finite energy cost of 2J. For any T 6= 0, both the parallel and antiparallel configuration at each of the red sites contribute to the statistical sum, destroying magnetic long-range order. The phase diagram is thus of type (b), and a percolation

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t

Fig. 3.

Red sites vs. red lines (shown in black) in a critical percolation cluster.

transition only occurs at exactly zero temperature. Since there are no thermal fluctuations, the critical behavior of this transition is identical to geometric percolation. Any T 6= 0 destroys the percolation critical behavior, instead the transition is in the universality class of the corresponding generic disordered classical magnet. 4. Diluted Quantum Magnets We now turn to the main topic, QPTs on percolating lattices. Generally, QPTs occur at zero temperature as functions of pressure, magnetic field or other nonthermal control parameters (for reviews, see, e.g., Refs. 11–14). One important aspect of these transitions is the so-called quantum-to-classical mapping. It arises because in quantum statistical mechanics the partition function does not factorize in potential and kinetic parts. Instead, it has to be formulated in terms of space and timedependent variables. As a result, (imaginary) time acts like an additional coordinate, and a QPT in d dimensions can be related to a classical transition in a higher dimension,a a fact we will be using repeatedly below. 4.1. Transverse-field Ising model The first example is a randomly diluted Ising model in a transverse magnetic field, given by the Hamiltonian X X X ˆ I = −J i Sîx − H i Sîz . (3) H i j Sîz Sˆjz − hx hi,ji

i

i

Sîz and Sîx are the z and x components of the the quantum spin-1/2 at site i; hx is the transverse field that controls the quantum fluctuations, and H is the field conjugate to the order parameter. The clean model (p = 0) is a paradigm for the a This

mapping is restricted to the thermodynamics only. Moreover some quantum transitions lead to extra complications such as Berry phases that do not have a classical counterpart.

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Fig. 4. Schematic T −hx −p phase diagram of a diluted quantum magnet . There is a multi-critical point at (T = 0, h∗x , pc ). The QPT across the dashed line is the topic of this paper.

study of QPTs: For hx J, the ground state is ferromagnetically ordered in zdirection while for hx J the quantum fluctuations due to the transverse field destroy the long-range order. The two phases are separated by a QPT at h x ∼ J.12 As in the classical case, we ask how the dilution influences the phase diagram (in the hx -p plane), i.e., is the phase diagram of type (a), (b), or (c) in Fig. 2? As before, (a) can be excluded because the infinite percolation cluster is a massive d-dimensional object for p < pc . To decide between (b) and (c), we adopt the “red-site” argument to the quantum case. Following the quantum-to-classical mapping, we have to consider an effective system in d space dimensions and one extra imaginary time dimension which becomes infinite for temperature T → 0. Crucially, the defect positions are time-independent. Instead of red sites we thus have “red lines” separating different pieces of the critical percolation cluster (right panel of Fig. 3). Configurations with different spin orientations on two such pieces now come with a infinite effective energy penalty and are suppressed. This suggests that magnetic long-range order can survive on the critical percolation cluster if the quantum fluctuations are not too strong, implying a phase diagram of type (c). This has been confirmed by simulation results not only for quantum Ising models but also Heisenberg magnets and quantum rotors.15–17 Combining the effects of thermal and quantum fluctuations, we obtain the T − hx − p phase diagram shown in Fig. 4. A diluted quantum magnet can thus undergo two nontrivial QPTs, separated by a multicritical point at (T = 0, h∗x , pc ). We are interested in the percolation transition at pc and hx < h∗x , i.e., the transition across the dashed line in Fig. 4. It was first investigated in detail by Senthil and Sachdev:18 Consider a single percolation cluster of s sites. For small transverse field hx < h∗x , all spins on the cluster are parallel. The cluster thus acts as a two-level system with an energy gap (inverse susceptibility) ∆s that depends exponentially on the cluster size ∆s ∼ χ−1 ∼ hx exp(−Bs) with B ∼ ln(J/hx ). (All other excitations have at s least energy J.) Since the cluster size s and its linear extension L are related via the fractal dimension s ∼ LDf , we obtain an unusual exponential relation between

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length and (inverse) time scales which is sometimes called activated scaling, ln(hx /∆s ) ∼ LDf .

(4)

The critical behavior of the total system can now be found by summing over all percolation clusters via the cluster size distribution (1). Let us first consider static quantities like magnetization or magnetic spatial correlation length. For hx < h∗x , magnetic long-range order survives on the infinite percolation cluster, while all finite-size clusters do not contribute. Thus, the total magnetization is proportional to the number of sites in the infinite cluster, m ∼ P∞ ∼ (pc − p)βc for p < pc . The magnetic order parameter exponent β is thus identical to that of geometric percolation. A similar argument can be made for the magnetic correlation length ξ: For hx < h∗x , all spins on a cluster are correlated, but the correlations cannot extend beyond the cluster size, thus ξ ∼ ξc ∼ |p − pc |−νc , and the magnetic correlation length exponent is identical to the geometric one, too. In contrast, quantities involving quantum dynamics behave nonclassically. For instance, the dependence of the magnetization on the ordering field H takes the scaling form (5) m(p − pc , H) = b−βc /νc m (p − pc )b1/νc , ln(H)b−Df , with b being an arbitrary scale factor. At the percolation threshold p = pc , this gives the unconventional relation m ∼ [ln(H)]2−τc . For p 6= pc , the transition is accompanied by strong power-law quantum Griffiths effects.6,18

4.2. Bilayer quantum Heisenberg magnet This subsection is devoted to diluted Heisenberg magnets. Specifically, we consider a dimer-diluted bilayer quantum Heisenberg antiferromagnet with the Hamiltonian X X ˆ H = Jk î,1 · S ˆ i,2 , î,a · S ˆ j,a + J⊥ H i S (6) i j S hi,ji a=1,2

i

where Sˆj,a is the spin operator at site j in layer a = 1 or 2. The clean system (p = 0) undergoes a QPT between a paramagnetic and an antiferromagnetic phase as a function of the ratio J⊥ /Jk between the inter-layer coupling and the in-plane interaction. For J⊥ Jk , the corresponding spins in the two layers form a singlet which is magnetically inert. Thus, there is no long-range order. In contrast, for Jk J⊥ , each layer orders antiferromagnetically, and the weak inter-layer coupling establishes antiferromagnetic order between the layers. The phase diagram of the dimer-diluted system has been determined by Sandvik16 and Vajk and Greven;19 it is shown in Fig. 5. In agreement with the general arguments given in the last subsection, long-range order can survive at p = pc giving rise to a nontrivial percolation QPT (across the short vertical line in Fig. 5). To study this transition, we first map the low-energy physics of (6) onto a quantum rotor model with the action12 Z X T XX i ωn2 Si (ωn )Si (−ωn ) . (7) A = dτ Jk i j Si (τ ) · Sj (τ ) + g i n hiji

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J⊥ / J||

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1.5

Jz

1 0.5 0

0

0.1

0.2

0.3

0.4

p Fig. 5. Phase diagram of the dimer-diluted bilayer Heisenberg antiferromagnet (after Refs. 16,19). Inset: Sketch of the system.

Here, each rotor variable Si (τ ) (a unit vector at site i and imaginary time τ ), describes a dimer of corresponding spins in the two layers. ωn is a Matsubara frequency, and the parameter g is related to the ratio J⊥ /Jk of the quantum Hamiltonian (6). Our approach20 is the same as in Sec. 4.1, we first consider a single percolation cluster of size s and then sum over all clusters by means of the cluster size distribution (1). For small g, all rotors on the cluster are correlated but collectively fluctuate in time. Thus, each cluster acts as a (0+1)-dimensional rotor model with magnetic moment s. Its low-energy properties can be easily found by a renormalization group calculation or dimensional analysis, leading to a scaling form of the free energy Fs (g, H, T ) = (g/s)Φ Hs2 /g, T s/g (8)

as a function of g, T , and magnetic field H. Here, Φ is a universal scaling function. Eq. (8) implies that the thermodynamics of a quantum spin cluster is more singular in its size s than that of a classical cluster. In particular, classically, the magnetic susceptibility increases like χcs ∼ s2 while in our quantum model at T = 0, it increases more strongly, χs ∼ s3 . The dynamical critical exponent can be obtained by relating the gap ∆ to the linear size L of the cluster via χs ∼ s2 /∆, giving ∆ ∼ s−1 ∼ L−Df . Thus, the dynamical exponent at the percolation QPT is z = Df . The total free energy is obtained by summing (8) over all clusters. This gives rise to the general scaling scenario 2 − α = (d + z) ν ,

β = (d − Df ) ν ,

γ = (2Df − d + z) ν ,

δ = (Df + z)/(d − Df ) ,

2 − η = 2Df − d + z .

(9) (10) (11) (12) (13)

All critical exponents are completely determined by two geometric percolation exponents (say Df and ν = νc ) together with the dynamical exponent z which contains

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242 Table 1. Critical exponents of the geometric and quantum percolation transition in two and three dimensions.20 exponent α β γ δ ν η z

classical −2/3 5/36 43/18 91/5 4/3 5/24 -

2d quantum −115/36 5/36 59/12 182/5 4/3 −27/16 91/48

classical −0.62 0.417 1.79 5.38 0.875 −0.06 -

3d quantum −2.83 0.417 4.02 10.76 0.875 −2.59 2.53

the information on the quantum fluctuations. Thus, α, γ, δ, and η are modified by the quantum dynamics while β and ν are unchanged. The resulting exponent values are shown in Table 1. We have recently confirmed the 2d results by performing large-scale Monte-Carlo simulations of the action (7).17,21 Let us note that the behavior of site-diluted (rather than dimer-diluted) Heisenberg antiferromagnets is more complicated because the effective action contains Berry phases. Recent computer simulations22 suggest that z ≈ 1.5Df to 2Df in this case. 4.3. Percolation and dissipation In many real systems, magnetic degrees of freedom are coupled to a dissipative environment of “heat bath modes” (e.g., electronic degrees of freedom in a metal or nuclear spins in molecular magnet). In this subsection, we study the influence of Ohmic dissipation on a percolation QPT.23 To this end we add baths of harmonic oscillators to the diluted transverse-field Ising model of Sec. 4.1, X 1 † z † ˆ ˆ ˆ (14) H = HI + i νi,n ai,n ai,n + λi,n Si (ai,n + ai,n ) , 2 i,n where ai,n and a†i,n are the annihilation and creation operators of the n-th oscillator coupled to spin i; νi,n is its frequency, and λi,n is the coupling constant. All baths P have the same spectral function E(ω) = π n λ2i,n δ(ω − νi,n )/νi,n = 2παωe−ω/ωc with α the dimensionless dissipation strength and ωc the cutoff energy. Following our general approach we first consider a single percolation cluster of size s. Without dissipation, its low-energy properties are described by a quantummechanical two-level system (see Sec. 4.1). In the presence of the heat baths, the cluster therefore behaves as a dissipative two-level system with effective dissipation strength sα. The physics of this problem is very rich, it has been reviewed, e.g., in Ref. 24. For our purposes, the most important aspect is that with increasing dissipation strength, the (Ohmic) dissipative two-level system undergoes a phase transition from a fluctuating phase at sα < 1 to a localized (frozen) phase at sα > 1. As a result, for any given microscopic dissipation strength α, the total diluted lattice consists of a mixture of large frozen clusters that act as classical superspins and smaller clusters that behave quantum mechanically down to the

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Fig. 6. Schematic ground state phase diagrams of the diluted transverse-field Ising magnet without (a) and with (b) dissipation. CSPM is the classical superparamagnetic phase (after Ref. 23).

lowest temperatures. The resulting phase diagram23 of the of the dissipative diluted transverse-field Ising model is shown in Fig. 6. The behavior of observables close to the percolation transition can be obtained by summing the results for the dissipative two-level system over the cluster size distribution (1). The total magnetization has 3 parts: The infinite percolation cluster, if any, contributes m∞ ∼ P∞ ∼ (pc − p)βc . The frozen finite size clusters individually have nonzero magnetization, but they do not align in the absence of an ordering field. Finally, the small fluctuating clusters have vanishing magnetization. The interplay between these 3 contributions and an ordering field leads to exotic hysteresis phenomena.23 The low-temperature susceptibility is dominated by the frozen clusters and behaves classically, χ ∼ |p − pc |γc /T . In contrast, the specific heat is determined by quantum fluctuations giving C ∼ 1/ ln2 (hx /T ). 5. Conclusion In summary, we have discussed the interplay of geometric, thermal and quantum fluctuations at the percolation threshold. While thermal fluctuations immediately destroy magnetic long-range order on the critical percolation cluster, the effects of quantum fluctuations are more subtle. Generically, magnetic long-range order on the critical percolation cluster can survive a finite amount of quantum fluctuations. This gives rise to a nontrivial percolation QPT and a multicritical point separating it from the generic “disordered” transition at p < pc . We have discussed three examples of such percolation QPTs in quantum Ising and Heisenberg magnets with and without dissipation. In all cases, the critical behavior is different from classical (geometric) percolation, but it can be expressed in terms of the geometric percolation critical exponents. Percolation transitions are thus among the very few examples of QPTs with exactly known exponent values in two dimensions. This is caused by the fact that at our percolation transitions, the criticality is due to the geometric criticality of the underlying diluted lattice. However, the quantum fluctuations “go along for the ride” and modify the behavior of all quantities involving dynamic correlations. In the quantum Heisenberg case (Sec. 4.2), this leads to new critical expo-

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nents while the overall power-law scaling scenario is still valid. In contrast, for the transverse-field Ising model (Sec. 4.1), the dynamical scaling is of activated (exponential) rather than the usual power-law type. Finally, in the presence of dissipation (Sec. 4.3), the singularities in the dynamics become even stronger such that individual finite-size clusters can undergo the phase transition independently from the bulk. This leads to a novel superparamagnetic classical cluster phase. Note that these different cases agree with a general classification of phase transitions in the presence of disorder6 based on the effective dimensionality of the defects. Acknowledgments We gratefully acknowledge discussions with M. Greven, S. Haas, H. Rieger, A. Sandvik, J. Schmalian, and M. Vojta. Parts of this work have been performed at the Aspen Center for Physics and the Kavli Institute for Theoretical Physics, Santa Barbara. This work has been supported by the NSF under grant no. DMR-0339147, by Research Corporation, and by the University of Missouri Research Board. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

M. Thill and D. A. Huse, Physica A 214, p. 321 (1995). H. Rieger and A. P. Young, Phys. Rev. B 54, p. 3328 (1996). D. S. Fisher, Phys. Rev. Lett. 69, p. 534 (1992). D. S. Fisher, Phys. Rev. B 51, p. 6411 (1995). T. Vojta, Phys. Rev. Lett. 90, p. 107202 (2003). T. Vojta, J. Phys. A 39, p. R143 (2006). D. Stauffer and A. Aharony, Introduction to Percolation Theory (CRC Press, Boca Raton, 1991). T. K. Bergstresser, J. Phys. C 10, p. 3381 (1977). M. J. Stephen and G. S. Grest, Phys. Rev. Lett. 38, p. 567 (1977). Y. Gefen, B. B. Mandelbrot and A. Aharony, Phys. Rev. Lett. 45, p. 855 (1980). S. L. Sondhi, S. M. Girvin, J. P. Carini and D. Shahar, Rev. Mod. Phys. 69, p. 315 (1997). S. Sachdev, Quantum phase transitions (Cambridge University Press, Cambridge, 1999). T. Vojta, Ann. Phys. (Leipzig) 9, p. 403 (2000). D. Belitz, T. R. Kirkpatrick and T. Vojta, Rev. Mod. Phys. 77, p. 579 (2005). A. B. Harris, J. Phys. C 7, p. 3082 (1974). A. W. Sandvik, Phys. Rev. Lett. 89, p. 177201 (2002). T. Vojta and R. Sknepnek, Phys. Rev. B. 74, p. 094415 (2006). T. Senthil and S. Sachdev, Phys. Rev. Lett. 77, p. 5292 (1996). O. P. Vajk and M. Greven, Phys. Rev. Lett. 89, p. 177202 (2002). T. Vojta and J. Schmalian, Phys. Rev. Lett. 95, p. 237206 (2005). R. Sknepnek, T. Vojta and M. Vojta, Phys. Rev. Lett. 93, p. 097201 (2004). L. Wang and A. W. Sandvik, Phys. Rev. Lett. 97, p. 117204 (2006). J. A. Hoyos and T. Vojta, Phys. Rev. B 74, p. 140401(R) (2006). A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg and W. Zwerger, Rev. Mod. Phys. 59, p. 1 (1987).

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GROUND-STATE PROPERTIES OF A HOMOGENEOUS 2D SYSTEM OF BOSONS WITH DIPOLAR INTERACTIONS G. E. ASTRAKHARCHIKa , J. BORONATa , J. CASULLERASa , I. L. KURBAKOVb , and YU. E. LOZOVIKb a

Departament de F´ısica i Enginyeria Nuclear, Campus Nord B4-B5, Universitat Polit` ecnica de Catalunya, E-08034 Barcelona, Spain b Institute of Spectroscopy, 142190 Troitsk, Moscow region, Russia The ground-state phase properties of a two-dimensional Bose system with dipole-dipole interactions is studied by means of quantum Monte Carlo techniques. Limitations of mean-field theory in a two-dimensional geometry are discussed. A quantum phase transition from gas to solid is found. The crystal is tested for the existence of a supersolid in the vicinity of the phase transition. The mesoscopic analogue of the off-diagonal longrange order is shown in the one-body density matrix in a finite-size crystal. A non-zero superfluid fraction is found in a finite-size crystal, the signal increasing dramatically in presence of vacancies. Keywords: Dipoles; supersolid; condensate fraction; equation of state.

1. Model and Methods We study the main ground-state properties of a two-dimensional (2D) system of bosons with dipolar interaction. We consider a polarized system and assume that dipolar moments are oriented perpendicularly to the 2D plane. This assures that the interaction potential Vint (r) = Cdd /|r|3 is always repulsive and there are no instabilities caused by dipolar attraction. The following model Hamiltonian is used to describe the system: N N 2 X Cdd X 1 ˆ =−~ ∇2i + , H 2m i=1 4π i<j |ri − rj |3

(1)

where m is the dipolar mass and N the number of dipoles. The properties of a homogeneous system are governed by one characteristic parameter, the dimensionless density nr02 , where the characteristic length r0 is proportional to the interaction strength: r0 = mCdd /4π~2 . Deeply in the dilute regime one expects that a short-range interaction potential (like the dipolar one) can be described by only one parameter, namely, the s-wave scattering length a. Parameters r0 and a are directly related: a = e2γ r0 = 3.17...r0 . We perform a numerical study of the ground-state properties of this system using quantum Monte Carlo methods. Firstly, using the variational Monte Carlo

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(VMC) method it is possible to evaluate multidimensional averages over the trial wavefunction ψT . In the calculation of the energy and superfluid fraction nS /n, the variational parameters in ψT are chosen such that they minimize the variational energy. In the calculation of the one-body density matrix, we optimize the parameters so that the difference between variational and mixed estimators is minimal. Secondly, we use the diffusion Monte Carlo (DMC) method based on solving Schr¨ odinger equation in imaginary time at T = 0. The DMC method permits to find the ground state energy E of a bosonic system exactly (in statistical sense). Also the superfluid density nS /n and local quantities (e.g. g2 (z), Sk , etc.) can be found in a “pure” (non-depending on the choice of the trial w.f.) way. An extrapolation procedure can be used for predictions of non-local quantities (e.g. g1 (z), nk , etc.).

2. Trial Wave Function We construct the trial wave function (w.f.) in the following form:

ψT (r1 , ..., rN ) =

N Y i<j

f2 (|ri − rj |) ×

M Y

k=1

N X l=1

exp{−α(rl −

rklatt. )2 }

!

(2)

where ri , i = 1, N, are the particle coordinates and rklatt. , k = 1, M, are the coordinates of triangular lattice sites. The trial w.f. (2) is symmetric under exchange of any two particles. The two-body Jastrow term f2 (r) is chosen1,2 at short distances as a solution of the 2-body scattering problem at zero energy. At large distances, the functional form of the hydrodynamic solution is used.3 Thus, f2 (r) accounts for pair-collisions relevant for short distances and collective behavior (sound) at large distances. One should note that ψT is not of a Nosanow-Jastrow type, as by moving one particle all M terms in the product in (2) are changed, thus introducing a global change (i.e. depending as well on coordinates of other particles), so that this term is not a one-body term, but rather a many-body term. Another feature of this w.f. is that the number of particles N can be different from the number of lattice sites M making it suitable for studying a crystal with vacancies. The parameter α describes particle localization close to lattice sites. The typical dependence (at sufficiently large density, nr02 & 10) of the variational energy on the parameter α shows two minima. The position of the first minimum is α = 0. In this case, the translational invariance is preserved and the density profile is flat. This minimum corresponds to a gas/liquid state. In the second minimum, α is finite and translational invariance is broken. This minimum corresponds to a solid state with a density profile that has crystal symmetry. Thus, with the single trial w.f. (2) and different variational parameters we are able to describe distinct phases.

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3. Results 3.1. Quantum phase transition The ground-state phase at small densities corresponds to a gas, the solid being metastable. On the contrary, at large densities the solid state is energetically preferable. The critical density of the quantum phase transition nc r02 = 290(30) was obtained by constructing fits to the energy of the gas and solid phases.2 This estimation of the critical density is in agreement with a Path Integral Monte Carlo calculation4 done at low finite temperature nc r02 = 320(140). A Green’s Function Monte Carlo calculation5 provided a slightly lower critical density nc r02 = 230(20), but in this case a discrete model was used and, by decreasing the filling factor, a small increase in the critical value was found. Fig. 1(a) shows the pair distribution function g2 (r) = hΨ† (0)Ψ† (r)Ψ(r)Ψ(0)i/n2 in the gas phase in a wide range of densities. At the largest density (close to the phase transition) there is a well pronounced first peak followed by a number of well visible oscillations. This is a manifestation of strong correlations present in the system close to the point of a quantum phase transition. As the density is lowered the height of the peak is decreased and eventually it disappears, leading to a smooth behavior without any visible oscillations. This smooth behavior is characteristic for weakly-interacting Bose systems.

3.2. Dilute regime In the dilute regime one expects the mean-field (MF) theory to be applicable. As was rigorously derived in Ref. 6, the 2D Gross–Pitaevskii equation (GPE) has a coupling constant g2D = 4π~2 /m| ln(na2 )| dependent on the density. This leads to a logarithmic dependence of the mean-field energy on the density 7 1 2π~2 EM F = g2D n = N 2 m| ln(na2 )|

(a)

(3)

(b)

Fig. 1. Correlation functions in gas phase: (a) Pair distribution function (b) One-body density matrix.

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It turns out that the mean-field contribution to the energy (3) is the only well established term. We perform a study of beyond mean-field terms. Comparison to numerical results for hard-disks8 show that for densities na2 . 10−6 the exact shape of the interaction potential is no longer important, and that the only relevant parameter is s-wave scattering length. The peculiarity of a two-dimensional system is that for such a small densities, 10−50 < na2 < 10−9 , there is a notable difference (of several percent) between MF-GPE and exact result. Moreover, to our knowledge there is no analytical theory able to reproduce correctly the energy in the whole region of the universal regime. Summarizing, the mean-field description has limitations (failure) in a two-dimensional system in the universal regime. The one-body density matrix g1 (r) = hΨ† (r)Ψ(0)i/n in the gas phase has a finite asymptotic value, as reported in Fig. 1b. In the thermodynamic limit (N → ∞) finite asymptotic values are manifestations of off-diagonal long-range order (ODLRO). The asymptotic value gives the condensate fraction. In the dilute regime almost all of the particles are condensed, but increasing the density the stronger interactions deplete the condensate and the condensate fraction drops down to 1% close to the phase transition point. An important question is what happens to the condensate as the system crystallizes. 3.3. Study of a supersolid There are several definitions of a supersolid: (1) Spatial order of a solid + finite superfluid density (2) Spatial order of a solid (broken-symmetry oscillations in diagonal element of OBDM) + off-diagonal long-range order in OBDM Generally, it is believed that both definitions are equivalent. Reduced dimensionality increases the role of quantum fluctuations. This makes a two-dimensional crystal a good candidate for having a supersolid. Notice that in a one-dimensional system quantum fluctuations destroy crystalline long-range diagonal order. A previous study for the presence a supersolid in a two-dimensional dipolar system is not conclusive. Low-temperature (PIMC) simulation4 shows that the gas phase is completely superfluid, while no superfluid fraction is found in crystal phase. Still, the presence of (a possible) supersolid can be masked by much smaller critical temperature in a crystal. A zero-temperature method was used with a symmetrized trial w.f. in Ref. 5. No conclusions were drawn for the presence/absence of a supersolid due to an insufficient overlap of the trial w.f. with the actual ground state. Fig. 2(a) shows the one-body density matrix g1 (r) in the crystal phase close to the phase transition. While the energy in a crystal is not sensitive to symmetrization of the wavefunction, it is crucial to symmetrize the w.f. in the calculation of g1 (r). Indeed, without symmetrization off-diagonal element g1 (r) decays exponentially fast

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(a)

(b)

nr02

Fig. 2. Solid phase, = 290: (a) one-body density matrix for different system sizes, solid linenon symmetrized w.f. (b) winding number, open symbols: N = M = 12 (circles), 16 (diamonds), 24 (squares); solid symbols M = 24, N = 23 (lower curve), M = 24; N = 22 (upper curve); dashed line - non symmetrized w.f; dotted line - diffusion constant of a free particle.

to zero, see thick line in Fig. 2(a). Using symmetrized w.f. we find instead a finite asymptotic value (of the order of 3 × 10−4 for N = 108 particles). There is a certain decay of the condensate fraction as the system size increases. Finite-size effects are very important in a 2D dipolar system. Indeed, the char√ acteristic dependence of the energy, OBDM limiting value, etc. is 1/ N instead of the typical law 1/N of short-range potentials, as can be seen from the tail correction of the potential energy. For this reason a proper study of the supersolid in the thermodynamic limit should be done. Here, we limit ourselves to some preliminary results for a finite-size system. The superfluid fraction at zero temperature corresponds to the slope of the diffusion coefficient D of the center of mass in imaginary time τ . Once again, symmetrization is crucial, otherwise an artificial zero slope is obtained (dashed line in Fig. 2b). Using properly symmetrized w.f. we find superfluid signal. The signal gets weaker as we increase the number of particles N (see open symbols in Fig. 2b). The trial w.f. (2) permits us to investigate the role of vacancies. A system with M = 24 lattice sites and 0; 1; 2 vacancies is studied. The superfluid density experiences dramatic effects in the presence of vacancies. The signal increases from ≈ 2% for 0 vacancies to ≈ 40% for 2 vacancies. 4. Conclusion To summarize, the diffusion Monte Carlo method was used to study the properties of a dipolar 2D Bose system at T = 0. The ground-state energy, pair distribution function, and one-body density matrix were calculated in a wide range of densities. The gas-solid quantum phase transition is found at a density nr02 = 290(30). Limitations (failure) of mean-field description were pointed out in the universal low-density regime. The existence of a mesoscopic analogue of the off-diagonal long-range order was shown in the one-body density matrix in a finite-size crystal close to the phase

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transition. A non-zero superfluid fraction was found in a finite-size crystal. The superfluid signal is dramatically increased in presence of vacancies. Acknowledgments This work has been partially supported by Grant FIS2005-04181 from DGI (Spain) and Grant No. 2005SGR-00779 from Generalitat de Catalunya. G.E.A. acknowledges post doctoral fellowship by MEC (Spain). References 1. G. E. Astrakharchik, J. Boronat, I. L. Kurbakov and Y. E. Lozovik, Phys. Rev. Lett. 98, p. 060405 (2007). 2. G. E. Astrakharchik, J. Boronat, J. Casulleras, I. L. Kurbakov and Y. E. Lozovik, Phys. Rev. A 75, p. 063630 (2007). 3. L. Reatto and G. V. Chester, Phys. Rev. 155, p. 88 (1967). 4. H. P. Buchler, E. Demler, M. Lukin, A. Micheli, N. Prokof’ev, G. Pupillo and P. Zoller, Phys. Rev. Lett. 98, p. 060404 (2007). 5. C. Mora, O. Parcollet and X. Waintal, cond-mat/0703620 . 6. E. H. Lieb, R. Seiringer and J. Yngvason, Commun. Math. Phys. 224, p. 17 (2001). 7. M. Schick, Phys. Rev. A 3, p. 1067 (1971). 8. S. Pilati, J. Boronat, J. Casulleras and S. Giorgini, Phys. Rev. A 71, p. 023605 (2005).

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LIQUID-SOLID TRANSITION IN BOSE SYSTEMS AT T = 0 K: ANALYTIC RESULTS ABOUT THE GROUND STATE WAVE FUNCTION E. VITALI∗ , D. E. GALLI and L. REATTO Dipartimento di Fisica, Universit` a degli Studi di Milano, via Celoria 16, 20133 Milano, Italy ∗ E-mail: [email protected] In this article we explore analytically the zero-temperature physics of a collection of structureless and spinless interacting bosons, focusing on the transformation properties of the ground state wave function under translations. The properties of this wave function may give an insight into the physical mechanism which gives rise to the liquid-solid transition taking place in samples of 4 He atoms at T = 0K. In the first part we show that, whenever the number of particles is finite, the ground state is translationally invariant at any density. In the thermodynamic limit, when the density is high enough, the extensivity of the Bragg peaks in the static structure factor S(~ q ) reveals a degeneracy of the ground state: there exist states with periodic local density modulations with the same energy as the translationally invariant ground state, and this is a scenario of translational symmetry breaking. Keywords: Phase-transitions; symmetry-breaking; supersolid.

1. Introduction The striking low–temperature properties of samples of 4 He, which seem to exhibit quantum coherence phenomena also in the crystalline phase,1 make very important the study of the properties of the ground state wave function. Since the strong interaction among the atoms rules out any perturbative approach and makes necessary to build up variational models, it is very interesting to investigate exact properties of the ground state wave function to improve the variational description. When dealing with the solid phase of a 4 He sample, it is an open problem whether one should use a wave function translationally invariant, like a Shadow wave function,2 in which crystallization is spontaneously induced by inter-particles correlations, or a wave function which explicitly breaks translational symmetry, like a Jastrow-Nosanow3 wave function. In this work we explore analytically the transformation properties under translations of the ground state wave function, starting from the Hamiltonian of N interacting bosons, and investigating the phase-transition from liquid to solid state, that takes place in 4 He at T = 0 K as the pressure exceeds 25 atm.

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2. The Model of the System As it is usual in the realm of low-temperature condensed-matter physics, we model a sample of 4 He atoms as a collection of N structureless and spinless bosons, interacting through a two-body potential v(~r). In order to give a mathematically meaningful statistical mechanical description of the system, it is necessary to confine the particles in a bounded region Ω ⊂ R3 . Since we study the bulk properties of the sample, we are not interested in introducing a physical environment, in the simplest way an infinite potential barrier, confining the Helium atoms; instead, we imagine the whole euclidean space R3 covered by an infinite set of identical copies of the confinement region Ω, and we use the recipes of periodic boundary conditions and extension of the interaction to the “images”, standard in the context of numerical simulations, so that, studying the degrees of freedom in the region Ω, we are indeed making a local description of a macroscopic sample. The natural Hilbert space for such a description is the symmetric functional ˆ : D ˆ ⊂ HBose → space HBose = ((L2 (Ω ))⊗N )sym , and the Hamiltonian operator H H HBose is: N N 2 X X ˆ =−~ v(~rî − ~rˆj ) ∇2i + H 2m i=1 i<j=1

(1)

where the domain DHˆ is made of (equivalence classes of) functions sufficiently smooth and which satisfy periodic boundary conditions on the walls of the confinement region, and in the definition of the potential energy operator we extend the interaction to the “images”. For technical reasons we choose the interaction potential to belong to the “Rollnik class”,4 a condition regarding an integral of v(~r), which is known to hold for the 4 He-4 He Aziz potential.5 It is proved in Ref. 6 that the Hamiltonian (1) defines a compact resolvent operator, its minimum eigenvalue is not-degenerate and the relative eigenfunction ˆ is real and strictly positive. Besides, for any τ > 0, e−τ H is trace class. 3. Translational Invariance In this way it results that the ground state wave function ψ0 (~r1 , ..., ~rN ), whenever the system is finite, is unique, real and almost everywhere positive. Theorem 3.1. The ground state wave function of the finite system ψ0 (~r1 , ..., ~rN ) is translationally invariant, that is ψ0 (~r1 − ~a, ..., ~rN − ~a) = ψ0 (~r1 , ..., ~rN ) for any ~a ∈ R3 , where, whenever a vector position falls outside the confinement region, we mean the periodic extension of the wave function. Proof. The geometrical property of the Hilbert space of the system ψ0 ∝ ˆ limτ →∞ e−(τ −E0 )H φ, which holds for any wave function φ ∈ HBose

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with non-zero overlap with the ground state wave function, hψ0 |φi ≡ R d~r1 ...d~rN ψ0 (~r1 , ..., ~rN )φ(~r1 , ..., ~rN ) 6= 0, together with the the known fact that Ω the convergence in L2 -sense implies the existence of a subsuccession punctually converging to the limit function, allows one to write ψ0 (~r1 − ~a, ..., ~rN − ~a) ∝ ˆ limτk →∞ (e−(τk −E0 )H φ)(~r1 − ~a, ..., ~rN − ~a). Now, the strict positivity of ψ0 makes possible to construct an element φ ∈ HBose strictly positive, such that φ(~r1 −~a, ..., ~rN −~a) = φ(~r1 , ..., ~rN ), with non-zero overlap hψ0 |φi with ψ0 ; we may take for example a Jastrow wave function. ˆ For such φ, it is very simple to prove that (e−(τk −E0 )H φ)(~r1 −~a, ldots, ~rN −~a) = ˆ ˆ (e−(τk −E0 )H φ)(~r1 , . . . , ~rN ), so that ψ0 (~r1 −~a, . . . , ~rN −~a) ∝ limτk →∞ (e−(τk −E0 )H φ) ˆ (~r1 − ~a, ..., ~rN − ~a) = limτk →∞ (e−(τk −E0 )H φ)(~r1 , ..., ~rN ) = ψ0 (~r1 , ..., ~rN ), which is the property of translational invariance of the ground state wave function. 4. Symmetry Breaking We have shown that, at any density, whenever the system is finite, the ground state wave function is translationally invariant. It is important to note that this result does not imply that the finite system is always in the liquid state, because the translational invariant wave function, depending on the thermodynamic state, can describe a crystalline phase as well.2 One could imagine the following physical picture: as the density increases, the role of the correlations becomes more and more important, so that, beyond a critical value, the atoms organize in a crystalline configuration, whose presence can be seen in the Bragg peaks of the static structure factor, while the center of mass wanders around so that there is indeed translational invariance. In the thermodynamic limit, the motion of the center of mass is forbidden by the super selection rules7 and a crystalline phase with broken translational symmetry becomes the stable state. Now we are going to give a mathematical description of this phase transition mechanism; let’s define the function ρ~q(~r) = ρav (1 + γ cos (~ q · ~r)), where ρav is the ˆ average particle density, and the real number Ec = minC (hψ|Hψi) which represents the minimum value for the energy in the class C of wave functions which give ρ~q(~r) as expectation value of the local density operator. The “Stiffness theorem”8 allows us to express the difference Ec −E0 , where E0 is the ground state energy in terms of the static limit χ(~ q , ω = 0) of the density-density response function8 of the system in the following way: δE(~ q ) ≡ Ec − E0 = −N 2

γ2 4V χ(~ q , ω = 0)

(2)

The inequality |χ(~ q , ω = 0)| ≥ (4N mS(~ q )2 )/(V ~2 |~ q |2 ), demonstrated in Ref. 9, which connects χ(~ q , ω = 0) and the static structure factor S(~ q ) = (hˆ ρ~q ρˆ−~qiψ0 )/N , allows us to write (2) in the form: |δE(~ q )| ≤ N γ 2

~2 |~ q |2 16mS(~ q )2

(3)

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From (3) we see, that, when the average density is high enough so that the static structure factor shows extensive Bragg peaks, revealing crystalline correlations among the atoms, in the thermodynamic limit states with periodic local density modulations have the same energy as the translationally invariant ground state. The wave vectors which govern the density modulations maximize the static structure factor and form the reciprocal lattice of the crystal. This is the typical scenario of symmetry-breaking. 5. Conclusion We have studied in a formal way the ground state of interacting many-bosons systems, focusing on the symmetry-breaking mechanism related to the zerotemperature liquid-solid transition. From this analysis one sees a physical picture in which many-body correlations spontaneously give rise to a crystalline order in the system, inducing a symmetry-breaking which manifests itself as one takes the thermodynamic limit; in our opinion, when building up a variational model of the ground state, the explicit introduction of the sites of a crystal lattice in the wave function would force the system into a crystalline configuration which may be far from the true equilibrium state of the system, achieved via inter-particle correlations. We think that it is advisable to construct variational wave functions which are translationally invariant, as the true ground state of the finite system, and let the crystalline phase spontaneously arise. The very intriguing possibility of a “supersolid state” in samples of 4 He atoms, makes very interesting to investigate, in future works, the possibility of both translational and U (1)-global-gauge symmetry breaking, in order to put light into the physical mechanisms which lead the non-classical rotational inertia effects recently observed.1 Acknowledgment One of us (E.V.) would like to thank Prof. Ludovico Lanz for useful discussions. References 1. 2. 3. 4. 5. 6. 7. 8. 9.

E. Kim, M.H.W. Chan, Nature 427, 225 (2004). S.A. Vitiello, K. Runge and M.H. Kalos, Phys. Rev. Lett. 60, 1970 (1988). L.H. Nosanow, Phys. Rev. Lett. 13, 270 (1964). M. Reed and B.Simon, Methods of Modern Mathematical Physics, Vol. II (Academic Press, New York, 1975). R.A. Aziz, F.R.W. McCourt, and C.C.K. Wong, Mol. Phys. 61, 1487 (1987). M. Reed and B.Simon, Methods of Modern Mathematical Physics, Vol. IV (Academic Press, New York, 1978). K.Huang, Statistical Mechanics, 2nd Edn. (Wiley, New York, 1987). G.F. Giuliani and G. Vignale, Quantum Theory of the Electron Liquid, Cambridge University Press, Cambridge (2005). F. Dalfovo and S. Stringari, Phys. Rev. B 46, 13991 (1992).

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THE EXACT RENORMALIZATION GROUP AND PAIRING IN MANY-FERMION SYSTEMS NIELS R. WALET School of Physics and Astronomy, University of Manchester, M13 9PL, UK ∗ E-mail: [email protected] We study the application of the exact renormalisation group to a many-body system consisting of fermions interacting through a short-range attractive force. This is modelled through an effective range expansion using an effective field theory inspired approach. We investigate a systematic description of many-body effects in such systems. Keywords: Exact renormalization group; pairing; BCS; BEC; cross-over.

1. Introduction The Cooper pairing mechanism, which takes place when there are attractive forces between fermions, even if they are too weak to produce two-body bound states, plays a crucial role in many areas of many-body physics. The ground-state wave function of the many-fermion system becomes qualitatively different by the pairing mechanism, which can be seen by the occurence of a phase transition with temperature. The order parameter for this transition is the energy gap in the fermionic spectrum. Within the paired ground state we can identify two extreme limits, depending on the interaction strength. The weak-coupling regime, defined by the absence of two-body bound states, manifests itself in Bardeen–Cooper–Schrieffer (BCS) superconductivity, while the strong-coupling regime, where there is a deeply bound state which approximates an elementary boson, corresponds to Bose–Einstein Condensation (BEC). Here we study pairing within a framework inspired by modern effective field theory (EFT). An EFT description of phenomena is intended to be generic, independent of complicated details of the underlying theory, depending only on the degrees of freedom and interactions relevant at the energy scale being considered. 1 A first approach to combining the notion of pairing with EFTs was given by Papenbrock and Bertsch,2 in nuclear physics. A more detailed analysis of the same problem, but cast in a slightly different language, was discussed at almost the same time in the condensed matter literature,3 and has a venerable history4 (see also5 ). There is also some closely related work by Weinberg.6 The main difference between Ref. [3] and Ref. [2] is that in the first reference the

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chemical potential µ is not linked to the Fermi momentum pF , so that the cross-over to negative µ, where BEC occurs, can be studied. In an attempt to find a way to include many-body physics beyond the mean field in an EFT-inspired approach we base our work on the use of the Exact Renormalisation Group (ERG). 2. The ERG The goal of the ERG approach is to determine the Legendre transform of the effective action, Γ[φc ] = W [J] − J · φc , where W is the usual partition function in the presence of an external source J.7 The quantity Γ, which reduces to the effective potential for homogeneous systems, can be used to generate 1PI Green’s functions for small fluctuations around the ground state. It is not easy to evaluate W [J] directly, but we can use an artificial renormalisation group flow to determine Γ.8 The running inherent in a renormalisation group is created by introducing an artificial gap in the energy spectrum for the fields which depends on a momentum scale k. Thus we define a different effective action for each k by integrating over high-momentum components of the fields only, with q > k. The RG trajectory then interpolates between the classical action of the underlying field theory (at large k), and the full effective action (at zero k).9 Clearly we must parametrise the effective action in such a way that we can describe the qualitatively different physics at different length scales. In our case this requires the possibility of the development of a gap in the fermion spectrum, which acts as a symmetry-breaking order parameter. This is zero for the classical action, but as we solve the ERG equations we move to the full effective action with spontaneously broken symmetry Here we shall study a system of fermions interacting through a four-fermion contact potential. We assume that the underlying field theory, in standard EFT fashion, describes the s-wave scattering of two fermions with phase shifts fully de4πa0 Thus negative scattering length termined by the scattering length, T = − M (1+ia 0 k) describes a system without a two-body bound state, with a pole on the positive imaginary axis, and positive scattering length a system with bound pairs, with a pole on the negative imaginary axis. The binding of such a pair gets deeper as a 0 gets smaller, while the limit a0 = ±∞ corresponds to a zero-energy bound state. Since we are interested in the appearance of a fermionic gap, we must have a dynamical field whose VEV describes that gap.6 At the upper end of the evolution (the starting value for large cut-off k) this field is irrelevant and is generated by a standard Hubbard–Stratonovich transformation of the four-fermion interaction. One would expect that, as we run the cut-off to smaller values, the dynamics of the boson field becomes more important, and will play a crucial role at low energies, but we shall show below that the truth is more complex. 3. RG Equation We treat a single species of fermion, as in neutron matter or an atomic BEC. Let φ be the boson field describing correlated pairs of fermions. We work with the

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following ansatz for Γ, †

†

Γ[ψ, ψ , φ, φ ] =

Z

Zm 2 d x φ (x) Zφ i∂t + ∇ φ(x) − U (φ, φ† ) 2m ZM 2 † ∇ ψ +ψ Zψ (i∂t + µ) + 2M i T i † † †T ψ σ2 ψφ − ψ σ2 ψ φ . −Zg 2 2 4

†

(1)

All parameters run with the scale of the regulator k; these include the wave-function renormalisations Zφ,ψ and the mass renormalisations Zm,M . As the cut-off increases the fermionic parameters ZM and Zψ go to one, while the bosonic Zφ and Zm go to zero. (M is the mass of the fermions in vacuum and m is naively chosen to have the value 2M , but its real role is only to make Zm dimensionless.) The fermion chemical potential µ should be determined from the relation ∂Γ/∂µ = n, where n is the number density of the conserved charge, i.e., baryon number for neutron matter. While we could fix µ at the starting point of the RG trajectory and keep it at that value (in which case particle number becomes a function of k), if we want to explore the full range of phenomena including BEC we must allow µ to go negative. In that case we choose µ to depend on k, keeping the number density fixed. The bosons carry twice the charge of a fermion, and couple to the chemical potential via the term 2µZφ φ† φ. This has been absorbed into the potential U . The fields φ have a non-standard normalisation, since we shall relate the VEV of φ to the gap via ∆2 = hφ† φi. This means that we have only a dimensionless coupling-constant renormalisation Zg for the boson-fermion coupling. This will be chosen to be fixed at unity here but in principle this is only correct in the absense of a gap. We define a “Fermi momentum” pF by the density, pF = (3π 2 n)1/3 . If we keep n fixed, pF will not change. In the renormalisation group evolution, we know that at high k we have the free action, which does not have a gap, the VEV of φ is zero. In this symmetric phase µ = p2F /(2M ), but at k = kcrit there will be a phase transition to a gapped phase, with spontaneously broken U (1) symmetry, which we also call the broken phase. We expand the potential U about its minimum to quartic order in the field, U (φ, φ† ) = u0 + u1 (φ† φ − ∆2 ) +

1 u2 (φ† φ − ∆2 )2 , 2

(2)

where the un are defined as the derivatives of U at its minimum, which occurs where φ† = φ = ∆. In the symmetric phase we typically have a minimum at the boundary, i.e., ∆ = 0, and u1 > 0. As we decrease k, at some point we expect u1 = 0, and the minimum moves away from ∆ = 0 for smaller k. In this condensed phase we expand

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around the k-dependent minimum, and we find that U (φ, φ† ) = u0 +

1 u2 (φ† φ − ∆2 )2 . 2

(3)

The renormalisation factors Zφ,ψ,m,M are evaluated in a bosonic back-ground corresponding to the minimum of the potential. Thus we neglect higher-order terms in the fields such as (φ† φ − ∆2 )[φ† i∂t φ − φi∂t φ† ]. The evolution equation for the ERG is quite straightforward, and can be written as9 i i h i i h (2) (2) ∂k Γ = − tr (∂k RB (k)) (ΓBB − RB )−1 + tr (∂k RF (k)) (ΓF F − Rf )−1 . (4) 2 2 (2)

Here ΓF F (BB) is the matrix containing second functional derivatives of the effective action with respect to the fermion (boson) fields and RB(F ) is a matrix containing the corresponding boson (fermion) regulators. This leads to a simple one-loop structure of the right-hand side of the ERG equations, and can be turned into equations for the parameters in the effective action discussed above. In the bosonic sector, the regulator is an additional quadratic term, proportional to φ† (x)φ(x0 ). It has the matrix structure RB (q, k) 0 RB (q, k) = , (5) 0 RB (q, k) where RB (q, k) is a scalar function. This provides an extra contribution to the single particle energies, which should suppress the contributions of states with momenta q . k. As k tends to zero we would recover the full effective action from the solution to the RG equation, if we had not made any truncations. This means that RB (k) should vanish as k → 0. To give all modes a large mass for large k, RB (q, k) should be large for momenta q . k. In the fermion case, in the symmetric phase our regulator should be positive for particle states (q > pF ) and negative for hole states (q < pF ), which reflects the fact that we have quantised relative to the vacuum rather than the Fermi sea, sgn(ZM (q) − µ) RF (q, pF , k) 0 RF (q, pF , k) = . 0 − sgn(ZM (q) − µ) RF (q, pF , k) (6) In the symmetric phase it should suppress the contributions of states with momenta near the Fermi surface, |q − pf | . k, but otherwise it should behave similarly to the bosonic cut-off function. If we ignore all the diagrams where bosons propagate we get the normal meanfield limit, provided that we do not expand the effective potential. Since we have a cut-off depence in our calculation, we find a slight generalisation of the results of Refs. [2,3]. We assume that in vacuum our theory reproduces the scattering length a0 , and that the subtraction in matter is identical to that in vacuum (see Ref. 1 and references therein for discussion of the vacuum problem). The full effective action at

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k = 0 coincides with the mean-field results quoted in these references. The evolution of the effective potential is quite straightforward, F

U (k, ∆, µ) =

Z

d3 q~ (2π)3

p 1 ∆2 − EF R (q, k) + ∆2 (q) − µ + 2 (q)

+

M ∆2 , (7) 4πa0

where we use the short hand (q) =

1 2 q , 2M

EF R (q, k) = (q) − µ + RF (q, pF , k) sgn((q) − µ).

(8)

In the limit k = 0 we can find a closed-form expression for the effective action. As 1/(pF a0 ) goes to −∞, the µ dependence becomes irrelevant (µ = F ), and we can obtain the standard result (see, e.g., Ref. [2]) 8 π ∆ = 2 F exp − . e 2pF |a0 |

(9)

If we work at constant density, minimizing U with respect to ∆ with the subsidiary condition ∂µ U = n, we get the results from Ref. [3], which can be turned into an implicit relation for the gap equation, 1 1 (y ))2/3 (−3π 2 (x20 + 1)1/4 P1/2 0 2 1 1 1 = − x0 (x20 + 1)1/4 P1/2 (y0 ) (∆/F )1/2 (y0 ) − P3/2 3 y0

∆/F =

(10)

1 a0 pF

(11)

with y0 = −x0 (x20 + 1)−1/2 . The interesting aspect of this result is that this relation is simpler than we would expect: the problem is goverened by three dimensionless parameters, a01pF , ∆/F and x0 = µ/F , and we would normally expect any one to depend on both other parameters.

4. Evolution Equations We now study the evolution equations. For the effective potential these can be obtained assuming small fluctuations around a constant background field. For the wave function renormalisation factor, Zφ , we need to consider a time-dependent background field. Taking φ(x) = φ0 + ηe−ip0 t ,

(12)

where η is a constant, we can get the evolution of Zφ from ∂ ∂ k Zφ = ∂p0

δ2 ∂k Γ δηδη †

η=0

. p0 =0

(13)

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The evolution equations in the symmetric phase are Z 1 d3 q~ 1 ∂ k u1 = − ∂k RF 3 2Zψ (2π) EF2 R Z Z 1 1 d3 q~ d3 q~ 3 u22 ∂ k u2 = − ∂ R + ∂k RB , k F 4 3 4Zψ (2π) EF R 2Zφ (2π)3 E (0)2 BR Z 1 d3 ~q 1 ∂k RF , ∂ k Zφ = − 3 2 (2π) EF3 R ∂k n = 0

(14) (15) (16) (17)

Here we define Zm 2 q + u1 + RB (q, k). (18) 2m The evolution equations in the broken phase are slightly more complicated, and can be found in Ref. [11]. To simplify our initial calculations we shall assume that (0)

EBR (q) =

Zψ (k) = 1,

ZM (k) = 1.

(19)

We solve these evolution equations starting from a large, fixed value of k, k = K, and evolve the solutions down from this value. This clearly requires some initial conditions. These are obtained by using the standard mean-field evolution from k = ∞ to k = K, i.e., by using Eq. (7) and related relations. By differentiating U F with respect to ∆2 at ∆ = 0 we find Z 1 d3 q~ 1 sgn((q) − µ) M . (20) + − u1 (K) = − 4πa0 2 (2π)3 EF R (q, K) (q)

The second term in the integral contains the divergent term in the free inverse T matrix; the first term corresponds to a finite density correction. There are some subtleties associated with the fact that we have only evaluated the partial derivative with respect to k. As explained above, in the broken phase we expand about a background that runs with k, since we wish to stay at the minimum of U , u1 = 0 and keep particle density n fixed as k decreases. This can only be done if we allow the parameters µ and ∆2 to evolve with k. This leads to dΓ = (∂k Γ + ∂∆2 Γ∂k ∆2 (k) + ∂µ Γ∂k µ)dk.

(21)

We can use this to derive evolution equations for ∆(k) and µ(k) by imposing du1 = dn = 0 in the broken phase. Unfortunately, this forces us to study the full dependence on ∆2 and µ of all quantities appearing in the effective action, and we cannot get explicit expressions for all of these. By performing analytic differentiation of the evolution equations, we can show that the evolution of ui involves ui+1 , etc. This is tackled by a closure approximation incorporating the exact fermionic solution. The underlying assumption is that bosonic contributions to the evolution of these quantities are small. We shall investigate this assumption later on, and find

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that it seems to be true generically. This can in principle be tested by truncating the hierarchy of equations further down the line, keeping more explicit bosonic evolutions, and checking for convergence with respect to the number of equations. We now integrate the resulting differential equations numerically. We use cutoff functions R based on a smoothed “θ” function, so that we can compare with analytical results for a sharp cut-off. This function decays quickly to zero for |q| > k, and is given by (q + k) (q − k) + erf − /(2 erf(1/σ)), (22) θ(q, k; σ) = erf − kσ kσ

where the normalization is chosen such that θ(0, k; σ) = 1. The cut-off functions R are defined in terms of these θs as k2 k2 θ(q − pF , k; σ), RB (q, k; σ) = θ(q, k; σ). (23) RF (q, k; pF , σ) = 2M 2m One point to note is that the fermionic cutoff is peaked about pF . In the symmetric phase this suppresses contributions from particles and holes around the Fermi surface, by giving a k-dependent energy gap. As noted above, in the broken phase the position of the “Fermi surface” is no longer at pF exactly, and indeed the Fermi surface may not be well defined. However, in that case a real gap has appeared and the role of the regulator is no longer crucial. 5. Results First of all, we checked the dependence of the results on the starting point K of the evolution, and found that as long as K > 5 fm−1 the dependence was undetectable. Similarly, we get the same numerical result for different values of the width parameter σ. A typical set of solutions for the evolution equations is given in Fig. 1. First, we see that the evolution of u1 and u2 in the presence of boson loops seems to coincide with the purely fermionic one until the point of transition, such that only one curve is actually visible. Below the transition the boson loops do have some effect, but surprisingly little, especially on the gap which only changes by less than 1% from the fermionic value. In the end the boson constants seem to be pushed to zero, which seems to be linked with our neglect of the particle-hole channels: the model with separable pairing is exactly solvable.12 The rate of running of ∆ due to boson loops depends in a subtle way on u−1 2 . It seems to be the reduction u2 due to bosonic loops which gives us the main correction to the gap. The running of Zφ seems to have less influence, but notice that in the area of interest Zφ is close to 1, which is the approximation made in generating the other curves. It is instructive to look at the result for the gap as a function of pF a0 . We have done four types of calculations, fermion loops only, inclusion of boson loops and on top of that have allowed Zφ to run, see Fig. 2 for a comparison. Finally we make the approximation Zm = Zφ . All of these contain the approximate closure discussed above.

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262 smooth, fixed width

smooth, fixed width

σ=0.1

σ=0.01

1

1

0.1

0.1

0.01

0.01

0.001

0.001 u1 µ u2 Zφ ∆

0.0001 1e-05 1e-06 0.0001

0.001

0.01

u1 µ u2 Zφ ∆

0.0001 1e-05 0.1

1

10

100

1e-06 0.0001

0.001

k smooth, k-scaled width 1

1 0.1

0.01

0.01

0.001

1e-05

100

0.001

0.01

10

100

10

100

u1 µ u2 Zφ ∆

0.0001 1e-05 0.1

1

10

100

1e-06 0.0001

0.001

k sharp, fixed width

0.01

0.1

1

k sharp, fixed width

σ=0.1

σ=0.01

1

1

0.1

0.1

0.01

0.01

0.001

1e-06 0.0001

10

0.001 u1 µ u2 Zφ ∆

0.0001

1e-05

1

σ=0.01

0.1

0.0001

0.1

k smooth, k-scaled width

σ=0.1

1e-06 0.0001

0.01

0.001 u1 µ u2 Zφ ∆ 0.001

0.0001 1e-05 0.01

0.1

k

1

10

100

1e-06 0.0001

u1 µ u2 Zφ ∆ 0.001

0.01

0.1

1

k

Fig. 1. The numerical solution to the evolution equations for the most complete approximation and 1/a0 = 0 fm. We have used three different types of cut-off functions. Note that there is very little difference in the final result. All calculations were started at K = 16 fm −1 .

We notice that there is little difference in the result for the gap, except at the lowest value of 1/(a0 pF ). This is somewhat surprising, since we naively expected bosons to dominate the large gap region, not the small gap one. The analysis of the equations shows that for large gaps the dominant solution to the differential equation is the fermion-loop contribution to the running of the gap, due to the fact that it goes like 1/u2 , and u2 is very small. Other terms are either suppressed by powers of the large gap or by powers of u2 . Actually, the only other important loop integral is the boson loop contribution to the running of the gap, but it is so much smaller than the fermionic contribution, that it seems to make little difference. On a second thought this is not a surprise: in this limit ∆ describes the binding energy

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MF

-1 (%)

1

0

∆/∆

April 25, 2008

-1

-2

-2

0

2

(pFa0)

4

6

-1

Fig. 2. Ratio of the gaps for various numerical solutions to the analytic mean-field result. Again pF = 1.37fm−1 . Red dots denote the result from a calculation with fermion loops only, the open circles (blue) show the results from adding bosonic loops, and the crosses (green) show the effect when Zφ runs as well.

of a pair of fermions, which is mainly determined by fermionic loops. The failure at small gaps is due to the non-analyticity of the fermionic effective action, which means that our truncation to u0 , u1 and u2 is no longer adequate. This exhibits itself in a divergence of the purely fermionic contribution to u2 (which behaves as ∆2 log(∆2 )). This also leads to the non-analytic behaviour of the gap at small a0 pF . Since we have only correctly described this non-analytic behaviour for calculations with fermionic loops, the results are very sensitive to small corrections due to the boson loops (especially to the non-analytic behaviour). Some of the results seem to indicate that the gap goes to zero as a power rather than an exponential, but a definite conclusion will have to await a more complete calculation. We have also investigated the dependence on pF , and found it to be weak apart from small values of 1/(a0 pF ), where it seems not too surprising that the nonanalytic behaviour of U is influenced by the density. 6. Discussion and Conclusion If we were to apply the current calculation to realistic neutron matter, we would find a gap of about F , of the order of 50 MeV, whereas most calculations with more realistic interactions obtain a value about 10 times smaller. There is a simple argument by Fayans.13 The argument can be given most succinct in the weak coupling limit, where the gap satisfies a generalisation of Eq. (9) ∆=

8 F exp (−π/2 cot(δ(pF ))) . e2

(24)

For N N scattering cot(δ(k)) increases relatively quickly with momentum, and the most natural extension to our work, namely inclusion of the effective range of the N N force, can capture that quite easily; for ordinary nuclear matter the reduction is substantial, and we get a gap of the right size.

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Clearly this shows that the introduction of an effective range in our calculations is quite important; there are many other steps which should be taken to improve our approach. The first change we would like to make is to include running of the fermion renormalisation constants and “Yukawa” coupling. Although we do not expect any significant changes in the results, such an improvement will make the whole approach more consistent. Among other things we would also like to include ph channels (RPA phonons) explicitly, and perform calculations at finite temperature. These subjects will be considered in future studies. We have presented the ERG-based study of the pairing phenomena in many fermion system. The dynamics of a system is described in terms of the effective action depending on some running scale k which can be thought as an infrared cutoff. We assume some ansatz for the effective action and derive a set of approximate flow equations for the effective couplings. These flow equations are solved numerically. It is shown that the system undergoes a phase transition at some critical value of a running scale. As a result of this transition the energy gap and Goldstone mode are developed. We demonstrate that the standard gap equation is recovered when the boson loop contribution is turned off. We found that the fermion loops give the main contribution at all scales, but we need to investigate the running of the coupling constant and the fermionic Z’s before we can make a definite statement. Acknowledgment This research was funded by the EPSRC. References 1. P. F. Bedaque and U. van Kolck, Ann. Rev. Nucl. Part. Sci. 52, p. 339 (2002) [nuclth/0203055] 2. T. Papenbrock and G. Bertsch, Phys. Rev. C 59, p. 2052 (1999). 3. M. Marani, F. Pistolesi and G. C. Strinati, Eur. Phys. J. B 1, p. 151 (1998) [condmat/9703160]. 4. D. M. Eagles, Phys. Rev. 186, p. 456 (1969); A. J. Leggett in Modern Trends in the Theory of Condensed Matter, A. Pekalski and J. Przystawa, Lect. Notes in Phys. 115 (Springer Verlag, Berlin, 1980), p. 13.; P. Nozieres and S. Schmitt-Rink,J. Low. Temp. Phys. 59, p. 159 (1985). 5. E. Babaev, Phys. Rev. B 63, p. 184514 (2001). 6. S. Weinberg, Nucl. Phys. B413, p. 567 (1994) [cond-mat/9306055]. 7. S. Weinberg, The quantum theory of fields, Vol. 2, Chapter 16 (Cambridge University Press, 1996). 8. K. G. Wilson and J. G. Kogut, Phys. Rept. 12C, p. 75 (1975). 9. J. Berges, N. Tetradis and C. Wetterich, Phys. Rept 363, p. 223 (2002) [hepph/0005122]. 10. B. Delamotte, D. Mouhanna and M. Tissier, Phys.Rev. B 69 (2004) 134413 [condmat/0309101]. 11. M.C. Birse, B. Krippa, N.R. Walet and J.A. McGovern, Phys. Lett. B 605, p. 287 (2005). 12. G. Ortiz and J. Dukelsky, Phys. Rev. A72,p. 043611 (2005). 13. S. A. Fayans and D. Zawischa, Int. J. Mod. Phys. B 15, p. 1684 (2001) [nuclth/0009034].

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THE SPIN-1/2 AND SPIN-1 QUANTUM J1 –J10 –J2 HEISENBERG MODELS ON THE SQUARE LATTICE R. F. BISHOP†∗ and P. H. Y. LI School of Physics and Astronomy, The University of Manchester, Manchester, M13 9PL, UK † E-mail: [email protected] R. DARRADI and J. RICHTER Institut f¨ ur Theoretische Physik, Otto-von-Guericke Universit¨ at Magdeburg, 39016 Magdeburg, Germany We study the J1 –J10 –J2 quantum spin model on the two-dimensional square lattice using the coupled cluster method. We compare and contrast the influence of the interchain coupling J10 on the zero-temperature phase diagrams for the two spin values s = 1/2 and s = 1. Our most important result for the s = 1/2 case is the predicted existence of a quantum triple point (QTP) at (J10 ≈ 0.60 ± 0.03, J2 ≈ 0.33 ± 0.02) when J1 = 1. Below the QTP (J10 /J1 . 0.60) we predict a second-order phase transition between the quasi-classical Néel and stripe-ordered phases, whereas the corresponding classical model, which contains only these two phases for all spin values s, yields a first-order transition. Above the QTP (J10 /J1 & 0.60) an intermediate disordered phase emerges, which has no classical counterpart. By contrast, the situation for s = 1 is qualitatively different. Instead of a QTP where three phases co-exist, we now predict a quantum tricritical point at (J10 ≈ 0.66 ± 0.03, J2 ≈ 0.35 ± 0.02) when J1 = 1, where a line of second-order phase transitions between the quasi-classical Néel and stripe-ordered phases (for J10 /J1 . 0.66) meets a line of first-order phase transitions between the same two states (for J10 /J1 & 0.66). Surprisingly, we find no evidence at all for any intermediate disordered phase in the s = 1 case. Keywords: J1 –J10 –J2 model; coupled cluster method; quantum phase transition; frustrated magnet; spin-lattice system; quantum triple point; quantum tricritical point; spinhalf model; spin-one model.

1. Introduction The frustrated Heisenberg antiferromagnet with nearest-neighbour J1 and competing next-nearest-neighbour J2 coupling (J1 –J2 model) has received renewed interest both theoretically (see Refs. [1–7] and references cited therein) and experimentally 8,9 due to the recent discovery or successful syntheses of such new magnetic materials, ∗ On

sabbatical leave during 2007–08 at William I. Fine Theoretical Physics Institute, School of Physics and Astronomy, University of Minnesota, 116 Church Street S.E., Minneapolis, MN 55455, USA.

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as the layered-oxide high-temperature superconductors, whose undoped precursors can be well described by the model. The interplay between frustration and quantum fluctuations in two-dimensional (2D) quantum spin systems can lead to rich and unusual phase scenarios between magnetically ordered semiclassical phases and novel quantum paramagnetic ground state phases (see Ref. [10] and references cited therein). An interesting generalization of the pure J1 –J2 model has also been introduced recently by Nersesyan and Tsvelik.11 They consider a 2D spatially anisotropic spin-1/2 J1 –J10 –J2 model on the square lattice, where the nearest-neighbour bonds have different strengths J1 and J10 in, say, the x (intrachain) and y (interchain) directions respectively. This model has been further studied by other groups using the exact diagonalization (ED) of small lattice samples of N ≤ 36 sites,12 and the continuum limit of the model.13 Both groups support the prediction by Nersesyan and Tsvelik11 of a resonating valence bond state for J2 = 0.5J10 J1 , and the limit of small interchain coupling extends along a curve nearly coincident with the line where the energy per spin is maximum. The model has also been studied by Moukouri14 using a two-step density-matrix renormalization group approach. Although spin problems are conceptually simple, they often exhibit rich and interesting phase diagrams due to the strong influence of quantum fluctuations in these strongly correlated systems. The strength of the quantum fluctuations can be tuned by varying either the anisotropy terms in the Hamiltonian15 or the spin quantum number s.16 Thus, lattice quantum spin problems maintain an important role in the study of quantum phase transitions. Very few calculations have been performed for the J1 –J10 –J2 model for the case of s = 1 up till now. It has, however, been studied using the two-step density-matrix renormalization group method, but only for the specific value of J10 /J1 = 0.2, and a second-order transition from a Néel phase to a disordered phase is observed with a spin gap.6 It has also been observed that quantum fluctuations can destabilize the ordered classical ground state (GS), even for large values of s, for large enough values of the frustration.1,5 Furthermore, it has been argued recently that the quantum phase transition between the semiclassical Néel phase and the quantum paramagnetic phase present in the 2D J1 –J2 model is not described by a Ginzburg-Landau type critical theory, but rather may exhibit a deconfined quantum critical point.17,18 The aim of this paper is to further the study of the J1 –J10 –J2 model by using the coupled cluster method (CCM). The CCM (and see Refs. [19–21] and references cited therein) is one of the most powerful and universally applicable techniques of quantum many-body theory. It has been applied successfully to calculate with high accuracy the ground- and excited-state properties of many lattice quantum spin systems (and see Refs. [7,21–25] and references cited therein). The CCM is appropriate for studying frustrated systems for which such other methods as quantum Monte Carlo techniques are limited by the infamous minus-sign problem, and exact diagonalization methods are restricted in practice to such small lattices that may be insensitive to the details of the often subtle phase order present.

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(b)

(c)

(d)

i,l

Fig. 1. (a) J1 –J10 –J2 model; — J1 ; - - - J10 ; - · - J2 ; (b) Néel state; (c) stripe state - columnar; and (d) stripe state - row. Arrows in (b), (c), and (d) represent spins situated on the sites of the square lattice (indicated by • in (a)).

2. The Model The J1 –J10 –J2 model is a general spin-s Heisenberg model on a square lattice with three kinds of exchange bonds, with strenth J1 along the row direction, J10 along the column direction, and J2 along the diagonals, as shown in Fig. 1. All exchanges are assumed positive here, and we set J1 = 1. The Hamiltonian of the model is described by X X H = J1 si,l · si+1,l + J10 si,l · si,l+1 i,l

+ J2

X i,l

i,l

(si,l · si+1,l+1 + si+1,l · si,l+1 ).

(1)

This model has two types of classical ground state, namely, the Néel (π, π) state and stripe states (columnar stripe (π, 0) and row stripe (0, π)), the spin orientations of which are shown in Figs. 1(b,c,d) respectively. There is clearly a symmetry under the interchange of rows and columns, J1 J10 , which implies that we need only consider the range of parameters with J10 < J1 . The ground state energies of the various classical states are given by cl EN´ eel = (−J1 − J10 + 2J2 )|s|2 , N cl Ecolumnar = (−J1 + J10 − 2J2 )|s|2 , N cl Erow = (J1 − J10 − 2J2 )|s|2 . (2) N We take J1 = 1 and J10 < 1. Clearly, from Eq. (2), the classical GS is then either the Néel state (if J10 > 2J2 ) or the stripe state (if J10 < 2J2 ). Hence, the (first-order) classical phase transition between the Néel and stripe (columnar) states occurs at J2c = 12 J10 , ∀J1 > J10 .

3. The Coupled Cluster Method Formalism The CCM formalism is now briefly described (and see Refs. [19–26] for further details). The starting point for any CCM calculation is to select a normalized model or reference state |Φi. It is often convenient to take a classical GS as a model state for

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CCM calculations of quantum spin systems. Accordingly our model states here are the Néel state and the stripe state. It is very convenient to treat each site on an equal footing, and in order to do so we perform a mathematical rotation of the local axes on each lattice site such that all spins in every reference state align along the negative z-axis. The Schr¨ odinger ground state ket and bra equations are H|Ψi = E|Ψi and ˜ ˜ respectively. The CCM parametrizes these exact quantum GS wave hΨ|H = EhΨ| ˜ = hΦ| ˜ Se ˜ −S . The correlation operators S functions in the forms |Ψi = eS |Φi and hΨ| P P + ˜ ˜ and S are expressed as S = I6=0 SI CI and S = 1 + I6=0 S˜I CI− , where C0+ ≡ 1, the unit operator, and CI+ ≡ (CI− )† is one of a complete set of multispin creation operators with respect to the model state (with CI− |Φi = 0 = hΦ|CI+ ), generically y + + + x written as CI+ ≡ s+ i1 si2 · · · sin , in terms of the spin-raising operators si ≡ si + si on lattice sites i. The ket- and bra-state correlation coefficients (SI , S˜I ) are calculated by requiring ¯ ≡ hΨ|H|Ψi ˜ the expectation value H to be a minimum with respect to each of them. This immediately yields the coupled set of equations hΦ|CI− e−S HeS |Φi = 0 ˜ −S HeS − E)C + |Φi = 0 ; ∀I 6= 0, which we solve for the correlation and hΦ|S(e I coefficients (SI , S˜I ). We may then calculate the GS energy from the relation E = hΦ|e−S HeS |Φi, and the GS staggered magnetization M from the relation M ≡ ˜ PN sz |Ψi which holds in the rotated spin coordinates. We note that we − N1 hΨ| i=1 i work from the outset in the N → ∞ limit. 4. Approximation Schemes The CCM formalism is exact if a complete set of multispin configurations {I} with respect to the model state is included in the calculation. However, it is necessary in practice to use approximation schemes to truncate the expansions in configurations ˜ For the case of s = 1/2 we employ here, {I} of the correlation operators S and S. 7,20–25 as in our previous work, the localized LSUBn scheme in which all possible multi-spin-flip correlations over different locales on the lattice defined by n or fewer contiguous lattice sites are retained. The numbers of such fundamental configurations (viz., those that are distinct under the symmetries of the Hamiltonian and of the model state |Φi) that are retained for the Néel and stripe states of the current model in various LSUBn approximations are shown in Table 1. We note next that the number of fundamental LSUBn configurations for s = 1 becomes appreciably higher than for s = 1/2, since each spin on each site i can now be flipped twice by the spin-raising operator s+ i . Thus, for the s = 1 model it is more practical to use the alternative SUBn–m scheme, where m is the size of the locale on the lattice and n is the maximum number of spin-flips. Hence all correlations involving up to n spin flips spanning a range of no more than m adjacent lattice sites are retained.21,26 In our case we set m = n, and hence employ the SUBn–n scheme. More generally, the LSUBm scheme is thus equivalent to the SUBn–m scheme for n = 2sm. Hence, LSUBm ≡ SUB2sm–m. For s = 1/2, LSUBn ≡ SUBn–n; whereas for s = 1, LSUBn ≡ SUB2n–n. The numbers of fundamental configurations retained

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269 Table 1. Numbers of fundamental configurations (] f.c.) for s = 1/2 and s = 1 in various CCM approximations. s = 1/2 Scheme

LSUB2 LSUB4 LSUB6 LSUB8 LSUB10

s=1

] f.c. Néel

stripe

2 13 146 2555 59124

1 9 106 1922 45825

Scheme

SUB2–2 SUB4–4 SUB6–6 SUB8–8 –

] f.c. Néel

stripe

2 28 744 35629 –

1 21 585 29411 –

at various SUBn–n levels for the s = 1 model are shown in Table 1. In order to solve the corresponding coupled sets of CCM bra- and ket-state equations we use parallel computing.27,28 5. Extrapolation Schemes In practice one needs to extrapolate the raw LSUBn or SUBn–n data to the n → ∞ limit. Based on our previous experience7,23,25 we use the following empirical threeparameter scaling laws for the extrapolations of the GS energy, E = a0 + a1 n−2 + a2 n−4 ,

(3)

and of the GS staggered magnetization for frustrated models, M = b0 + b1 n−ν ,

(4)

where the exponent ν is also a fitting parameter. We list below three fundamental rules, also based on our experience, as guidelines for the selection and extrapolation of the CCM raw data, using any approximation scheme. • Rule 1: As a fundamental rule of numerical fitting or numerical analysis, one should always have at least (n + 1) data points in order to have a robust and stable fit to any formula that contains n unknown parameters. This rule takes precedence over all other rules. • Rule 2: Whenever possible one should avoid using the lowest (e.g., LSUB2, SUB2-2) data points since such points are rather far from the large-n limit, unless it is necessary to do so to avoid breaking Rule 1. • Rule 3: If Rule 2 has been broken then some other careful consistency checks should also be performed. In our results below the LSUBn results for n = {4, 6, 8, 10} are extrapolated for s = 1/2, in order to preserve the above three rules, whereas the SUBn–n results for n = {2, 4, 6, 8} are extrapolated for s = 1, in each case using the schemes indicated above. For both the s = 1/2 and the s = 1 models we perform two separate sets

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Néel : J1’=0.2 J1’=0.4 J1’=0.6 J1’=0.8 J1’=1.0 stripe : J1’=0.2 J1’=0.4 J1’=0.6 J1’=0.8 J’=1.0 0.3 0.4 0.5

Néel : J1’=0.2 J1’=0.4 J1’=0.6 J1’=0.7 J1’=0.8 J1’=0.9 J1’=1.0 stripe : J1’=0.2 J1’=0.4 J1’=0.6 J1’=0.7 J1’=0.8 J1’=0.9 J1’=1.0

-1.8

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0

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(a)

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J2 Fig. 2.

Termination point

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(b)

Ground state energy per spin, E/N (with J1 = 1): (a) for s = 1/2; and (b) for s = 1.

of CCM calculations for given parameters (J1 ≡ 1, J10 , J2 ) based respectively on the Néel state and the stripe state as the model state |Φi. 6. Results and Discussion Figure 2 shows the GS energy per spin as a function of J2 for various values of J10 (all with J1 ≡ 1), extrapolated from both the s = 1/2 and s = 1 models from the raw CCM data as discussed above. Both the raw LSUBn data for the s = 1/2 model and the raw SUBn–n data for the s = 1 model terminate at some particular values. This occurs for the CCM curves based on both the Néel state and the stripe state as the model state |Φi. In all cases such a termination point arises due to the solutions of the CCM equations becoming complex at this point, beyond which there exist two branches of complex-conjugate solutions. In the region where the solution reflecting the true physical situation is real, there actually also exists another real solution. However, only the (shown) upper branch of these two solutions reflects the true physical situation, whereas the lower branch does not. The branch reflecting the true physical situation of the solutions is the one which becomes exact in some appropriate (e.g., perturbative) limit. This physical branch then meets the corresponding unphysical branch at some termination point beyond which no real solutions exist. The termination points shown in Fig. 2 are the extrapolated n → ∞ termination points and are evaluated using data only up to the highest level of the CCM approximation schemes used here, namely LSUB10 for the s = 1/2 model and SUB8–8 for the s = 1 model. The SUBn–n and LSUBn termination points are also reflections of phase transitions in the real system, as we discuss more fully below. We observe from Fig. 2 that for the case of the s = 1/2 model the two curves, based on the Néel and stripe model states, for a given value of J10 , cross (or, in the limit, meet) very smoothly near their maxima for all values of J10 . 0.6, at a value of J2 slightly larger than the classical transition point of 0.5J10 . This behaviour is indicative of a second-order quantum phase transition between these two phases, by contrast with the first-order classical transition from Eq. (2). Conversely, for values

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J10 & 0.6 the curves no longer cross at a physical value (viz., where the calculated staggered magnetization is positive), indicating the opening up of an intermediate quantum phase between the Néel and stripe phases. For the case of s = 1, the extrapolated GS energy curves of the Néel and stripe states again meet smoothly with the same slope for J10 . 0.66 ± 0.03. This behaviour is again indicative of a secondorder phase transition. By contrast, for J10 & 0.66 ± 0.03 the behaviour is typical of a first-order phase transition where the curves now cross with a discontinuity in the slope. Figure 2 clearly shows the distinct differences in the GS energy curves for the two models with s = 1/2 and s = 1. This different behaviour observed in the GS energy for the two models is reinforced by the GS staggered magnetization results discussed below. For the GS staggered magnetization for the s = 1/2 model we find that the extrapolation of Eq. (4) produces smooth and physically reasonable results, except for a very narrow anomalous “shoulder” region near the points where M vanishes for 0.6 . J10 . 0.75 for the Néel state. This critical regime is undoubtedly difficult to fit with the simple two-term scheme of Eq. (4). Our method for curing this problem and for stabilizing the curves is to make efficient use of the information we obtain in Eq. (4) to extract the exponent ν, and then to use that value to infer the next term in the series. We find, very gratifyingly, that the value for ν fitted to Eq. (4) turns out to be very close to 0.5 for all values of J10 and J2 except very close to the critical point. Therefore, we use the form of Eq. (5), (5) M = c0 + n−0.5 c1 + c2 n−1 . The use of Eq. (5) removes the anomalous shoulder. Henceforth, in all of the results we discuss, we use Eq. (5) for the staggered magnetization. We have also checked that for the s = 1/2 model the extrapolated results using the data sets with n = {2, 4, 6, 8, 10} and n = {4, 6, 8, 10} are very similar, thereby adding credence to the validity and stability of our results. Conversely, the results using the data set with n = {6, 8, 10} again display a minor spurious “shoulder” which is almost certainly due to violating our Rule 1. For the s = 1 model, no narrow anomalous “shoulder” region is observed in the raw SUBn–n results. We have also performed some vigorous tests in the extrapolation schemes for the staggered magnetization in this case. Our main finding is that Eq. (5) using the data set with n = {2, 4, 6, 8} is the most consistent in terms of both the GS energy meeting point and the staggered magnetization critical point, as discussed below. Figure 3 shows our extrapolated results for the GS staggered magnetization M for both models. The quantum phase transition or critical point marking the end of either the quantum Néel state or the quantum stripe state is determined by calculating the order parameter M for various values of J10 to obtain those values of J2 where M vanishes. However, as seen in Fig. 3, there also occur cases where the order parameters of the two states meet before the order parameter vanishing point. In these cases we take the meeting point to define the phase boundary between

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stripe

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0.3 0.6 0.2

J1’=0.2 J1’=0.4 J1’=0.6 J1’=0.8 J1’=1.0

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0 0

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M

M

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0 0

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(a) Fig. 3. s = 1.

0.6

0.8

1

J2

J2 (b)

Ground state staggered magnetization, M (with J1 = 1): (a) for s = 1/2; and (b) for

the quantum Néel and quantum stripe states. Thus, our definition of the quantum critical point is the point where there is an occurrence of a phase transition between the two states considered or where the order parameter vanishes, whichever occurs first. For the s = 1/2 model we note that M vanishes for both the quantum Néel and stripe phases at almost exactly the same critical value of J2 , for a given J10 , so long as J10 . 0.6. Conversely, for J10 & 0.6 there exists an intermediate region between the critical points at which M → 0 for these two phases. The order parameters M of both the Néel and the stripe phases vanish continuously both below and above the point J10 ≈ 0.60, as is again typical of second-order transitions. By contrast, we note the surprising result for the s = 1 model that the order parameter M goes to zero smoothly at the same point for both the quantum Néel and stripe phases with the same value of J10 , for all values of J10 . 0.66 ± 0.03, whereas the corresponding curves for the two phases meet at a nonzero value for higher values of J10 . Thus, in this regime we have behaviour typical of a secondorder phase transition between the quantum Néel and stripe phases. Furthermore, the transition occurs at a value of J2 very close to the classical transition point at J2 = 0.5J10 . Conversely, for values of J10 & 0.66 ± 0.03, the order parameters M of the two states meet at a finite value, as is typical of a first-order transition. We show in Fig. 4 the zero-temperature phase diagrams of both the spin-1/2 and spin-1 J1 –J10 –J2 models on the square lattice, as obtained from our extrapolated results for both the GS energy and the GS order parameter M . In the case of the spin-1/2 model our results provide clear and consistent evidence for a quantum triple point (QTP) at (J10 ≈ 0.60 ± 0.03, J2 ≈ 0.33 ± 0.02) for J1 = 1. For J10 . 0.60 there exist only the Néel and stripe phases, with a second-order transition between them, whereas for J10 & 0.60 there also exists an intermediate (disordered, paramagnetic) quantum phase, which requires further investigation. Although the nature of the intermediate phase is still under discussion, a valence-bond crystal phase seems

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273 M∆ ≡ M critial point c Emeet ≡ Energy meeting/crossing point of the Néel and stripe states

0.4

0.5

M∆ ≡ M critical point c Emeet ≡ Energy meeting/crossing point of the Néel and stripe states

0.4

QTP

QTCP

0.3

Néel

0.2

0.3 M∆ : Néel : M ≈ 0 M∆ : stripe : M ≈ 0

0.2

c

M∆ : Néel : M > 0

M∆ : Néel

c

M∆ : stripe : M > 0

c

0.1

0.1

M∆ : stripe

c

Emeet : 2nd order transition Emeet : 1st order transition

c

Emeet : 2nd order transition 0 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

J1’ (a)

Néel

→

stripe

↑

stripe

J2

0.5

J2

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1

J1’ (b)

Fig. 4. Ground state phase diagrams (with J1 = 1): (a) for s = 1/2, showing a quantum triple point (QTP); and (b) for s = 1, showing a quantum tricritical point (QTCP).

to be the most favoured from other investigations.2,17 On the other hand, another possibility for this intermediate phase is the resonating valence bond (RVB) phase.12 Other calculations on this spin-1/2 model12,13 differ predominantly by giving a QTP at (J10 = 0, J2 = 0) for J1 = 1. We believe that the difference arises essentially from the nature of the alternative methods used. For example, due to the small size of the lattices used, the ED calculations of Sindzingre12 might easily miss the longerrange correlations that become increasingly important the nearer one approaches the QTP. Unlike the s = 1/2 case there is no sign at all of any intermediate disordered phase for any value of the parameters J10 or J2 (for J1 = 1) for the case of s = 1. Hence, in this respect the quantum spin-1 model is much closer to the classical case, viz., the s → ∞ limit. However, unlike the classical case, there now appears to be a quantum tricritical point (QTCP) at (J10 ≈ 0.66 ± 0.03, J2 ≈ 0.35 ± 0.02) for J1 = 1, where a tricritical point is defined here to be a point at which a line of second-order phase transitions meets a line of first-order phase transitions. We note that the behaviour of both the order parameter (which goes to zero smoothly at the same point for both Néel and stripe phases below the QTCP, but which goes to a nonzero value above it) and the GS energy curves for the two phases (which meet smoothly with the same slope below the QTCP, but which cross with a discontinuity in slope above it) tell exactly the same story. In conclusion we note that two of the unique strengths of the CCM are its ability to deal with highly frustrated systems as easily as unfrustrated ones, and its use from the outset of infinite lattices. These, in turn, lead to its ability to yield accurate phase boundaries even near quantum triple and tricritical points. Our own results for the ground state energy and staggered magnetization provide a set of independent checks that lead us to believe that we have a self-consistent and coherent description of these extremely challenging spin-1/2 and spin-1 systems.

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References 1. P. Chandra and B. Doucot, Phys. Rev. B 38, 9335 (1988). 2. J. Richter, Phys. Rev. B 47, 5794 (1993); J. Richter, N.B. Ivanov, and K. Retzlaff, Europhys. Lett. 25, 545 (1994). 3. H.J. Schulz, T.A.L. Ziman, and D. Poilblanc, J. Phys. I 6, 675 (1996). 4. R.R.P. Singh, W.H. Zheng, J. Oitmaa, O.P. Sushkov and C.J. Hamer, Phys. Rev. Lett. 91, 017201 (2003). 5. F. Kr¨ uger and S. Scheidl, Europhysics Lett. 74, 896 (2006). 6. S. Moukouri, Phys. Lett. A352, 256 (2006). 7. D. Schmalfuß, R. Darradi, J. Richter, J. Schulenburg, and D. Ihle, Phys. Rev. Lett. 97, 157201 (2006). 8. R. Melzi, P. Carretta, A. Lascialfari, M. Mambrini, M. Troyer, P. Millet, and F. Mila, Phys. Rev. Lett. 85, 1318 (2000). 9. P. Carretta, N. Papinutto, C. B. Azzoni, M.C. Mozzati, E. Pavarini, S. Gonthier, and P. Millet, Phys. Rev. B 66, 094420 (2002). 10. J. Richter, J. Schulenburg, and A. Honecker, in Quantum Magnetism, edited by U. Schollw¨ ock, J. Richter, D.J.J. Farnell, and R.F. Bishop, Lecture Notes in Physics, Vol. 645, (Springer-Verlag, Berlin, 2004), pp. 85–153. 11. A.A. Nersesyan and A.M. Tsvelik, Phys. Rev. B 67, 024422 (2003). 12. P. Sindzingre, Phys. Rev. B 69, 094418 (2004). 13. O.A. Starykh, and L. Balents, Phys. Rev. Lett. 93, 127202 (2004). 14. S. Moukouri, J. Stat. Mech. P02002 (2006). 15. R. Darradi, J. Richter, and S.E. Kr¨ uger, J. Phys.: Condens. Matter 16, 2681 (2004). 16. R. Darradi, J. Richter, and D.J.J. Farnell, J. Phys.: Condens. Matter 17, 341 (2005). 17. J. Sirker, W.H. Zheng, O.P. Sushkov, and J. Oitmaa, Phys. Rev. B 73, 184420 (2006). 18. T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M.P.A. Fisher, Science 303, 1490 (2004); T. Senthil, L. Balents, S. Sachdev, A. Vishwanath, and M.P.A. Fisher, Phys. Rev. B 70, 144407 (2004). 19. R.F. Bishop, Theor. Chim. Acta 80, 95 (1991). 20. R.F. Bishop in Microscopic Quantum Many-Body Theories and Their Applications, edited by J. Navarro and A. Polls, Lecture Notes in Physics, Vol. 510 (Springer-Verlag, Berlin, 1998), pp. 1–70. 21. D.J.J. Farnell and R.F. Bishop, in Quantum Magnetism, edited by U. Schollw¨ ock, J. Richter, D.J.J. Farnell, and R.F. Bishop, Lecture Notes in Physics, Vol. 645 (SpringerVerlag, Berlin, 2004), pp. 307–348. 22. C. Zeng, D.J.J. Farnell, and R.F. Bishop, J. Stat. Phys. 90, 327 (1998). 23. S.E. Kr¨ uger, J. Richter, J. Schulenburg, D.J.J. Farnell, and R.F. Bishop, Phys. Rev. B 61, 14607 (2000). 24. D.J.J. Farnell, R.F. Bishop, and K.A. Gernoth, J. Stat. Phys. 108, 401 (2002). 25. R. Darradi, J. Richter, and D.J.J. Farnell, Phys. Rev. B 72, 104425 (2005). 26. D.J.J. Farnell, K.A. Gernoth, and R.F. Bishop, Phys. Rev. B 64, 172409 (2001). 27. We use the program package CCCM of D.J.J. Farnell and J. Schulenburg, see http://www-e.uni-magdeburg.de/jschulen/ccm/index.html. 28. D.J.J. Farnell, J. Schulenburg, J. Richter, and K.A. Gernoth, Phys. Rev. B 72, 172408 (2005).

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LIQUID-GAS PHASE TRANSITION IN NUCLEAR MATTER WITHIN A CORRELATED APPROACH A. RIOS∗ National Superconducting Cyclotron Laboratory, East Lansing, 48823-1 Michigan, USA ∗ E-mail: [email protected] A. POLLS and A. RAMOS Departament d’Estructura i Constituents de la Mat` eria, Universitat de Barcelona, Avda. Diagonal 647, E-08028 Barcelona, Spain ¨ H. MUTHER Institut f¨ ur Theoretische Physik, Universit¨ at T¨ ubingen, D-72076 T¨ ubingen, Germany The existence of a liquid-gas phase transition for hot nuclear systems close to saturation densities is an interesting prediction of finite temperature nuclear many-body theory. We have applied the realistic Self-Consistent Green’s Function’s (SCGF) method together with the Luttinger-Ward (LW) formalism to the study of the thermodynamical (TD) properties of infinite symmetric nuclear matter. We compare our results with those obtained within the Brueckner–Hartree–Fock (BHF) theory and find substantial differences. Keywords: Nuclear matter; many-body nuclear problem; liquid-gas phase transition.

1. Introduction Despite its potential applications in the understanding of some astrophysical scenarios (such as the thermal evolution of protoneutron stars) as well as in the description of intermediate energy heavy ion collisions, the study of the thermal behavior of nuclear systems within realistic many-body approaches has received a relatively scarce attention. In these systems, one can reach temperatures of the order of tenths of MeV, as large as nuclear energy scales, and therefore it is necessary to build calculation schemes within which one can study the interplay of the dynamical correlations introduced by the strong nuclear force and the thermal effects that influence the nuclear many-body wave function at such large temperatures. Contrary to what happens in nuclear systems, the thermal properties of other fermionic systems have been studied much more thoroughly. In particular, the available experimental data and the presence of non-analytical temperature dependences have triggered an important amount of theoretical studies concerning the thermal properties of liquid 3 He.1,2

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In the following, we study the thermodynamics of symmetric nuclear matter (SNM), an ideal, infinite, high-density system composed of the same amount of neutrons and protons interacting via the strong force. SNM is customarily used in the context of realistic nuclear many-body calculations because it has well-known single-particle wave functions (plane waves) and because it is expected to model approximately the interior of heavy nuclei. A realistic calculation dealing with nuclear matter should take into account that the short-range and tensor components of the nucleon-nucleon force modify so much the many-body wave function that it cannot be described in terms of mean-field Slater determinants. In addition, and to be consistent with the experimental evidence coming from (e, e0 p) experiments, one should consider fragmented single-particle states. The SCGF method within the ladder approximation includes both of these effects and therefore goes beyond the mean-field and the quasi-particle pictures. An important advantage of this formalism stems from the fact that it is based on well-founded microscopic grounds, i.e. the perturbation expansion of the propagator at finite temperature.3 This fact, together with the self-consistent procedure used in determining the one-body propagator, guarantees that our approximation is “conserving” and therefore fulfills TD consistency.

2. Thermodynamical Properties of Nuclear Matter The one-body properties of a many-body system can be obtained from the one-body propagator, G.4 The total energy of the system (a two-body operator for two-body interactions) can be computed from G too, via the Galitskii-Migdal-Koltun (GMK) sum-rule. And, in addition, the partition function of the system can be obtained from G with the help of the LW formalism.5 Within this formalism, one can compute a dynamical quasi-particle (DQ) entropy for the correlated system, which includes the effects of the fragmentation of the quasi-particle peak in the entropy in an approximate way.6 The free energy of the system can then be obtained from the difference of the GMK sum-rule energy and the DQ entropy, F = E GM K − T S DQ . The results obtained from the combination of the SCGF scheme and the LW formalism are displayed in Fig. 1(a), where we show the free energy per particle and the chemical potential as a function of the density for a temperature of T = 10 MeV. The results have been obtained with the CDBONN potential. We compare the SCGF results with those of an extension of the BHF formalism to finite temperatures. For the free-energy per particle, the SCGF scheme yields slightly more repulsive results than the BHF method. This difference is essentially due to the intermediate propagation of hole-hole pairs,3 which is lacking in the BHF calculation. To study the fulfillment of TD consistency, we should compare the values obtained for a quantity that can be computed either microscopically (from Green’s function theory) or macroscopically (from the TD properties of the system). We have chosen the chemical potential, which can be computed microscopically from the normalization of the momentum distributions to the density of the system,

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-20

1

SCGF pressure BHF pressure

F /A BHF µ ~BHF µ

F /A SCGF µ SCGF ~ µ

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SCGF

p [MeV fm ]

T=10 MeV

F/A, µ [MeV]

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0.3

0.4

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ρ [fm ] (a)

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0.1

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ρ [fm ] (b)

Fig. 1. (a) Free energies per particle (full lines) and microscopic chemical potentials µ ˜ (dotted lines) for the SCGF (circles) and BHF (diamonds) approaches. The macroscopic chemical potentials µ are also shown (dashed line for SCGF and dash-dotted line for BHF). (b) Pressure of symmetric nuclear matter within the SCGF (solid) and BHF (dashed) approaches.

P ˜), or macroscopically, from the derivatives of the free energy with ρ = k n(k, µ ∂F respect to density, µ = ∂N . While the differences between µ ˜ and µ for the BHF approach can be larger than 10 MeV, the SCGF results fulfill TD consistency with less than an MeV of error in the full density range. This shows that the numerical implementation of the ladder approximation by means of the SCGF scheme leads to TD consistent results and therefore preserves the well-known conservation properties of the ladder approximation. 3. Liquid-gas Phase Transition The pressure of SNM as a function of the density for both the SCGF and the BHF approaches at T = 10 MeV are shown in Fig. 1(b). These have been obtained from the TD relation p = ρ(F/A−µ). As intuitively expected, the repulsive effect of holes in the free energy is translated in a larger pressure of this approach with respect to the BHF at intermediate densities. However, for both approaches the pressure decreases with the density in a given range, signaling a mechanical instability of the system. The range of densities and temperatures where this happens defines the so-called spinodal region. This mechanical instability is associated to a first order phase transition, in which the system condenses from a gas phase at low densities to a liquid phase at higher densities. This phase transition for infinite nuclear matter should have its counterpart in finite systems. As a matter of fact, some measurements in heavy ion collisions at intermediate energies can be interpreted in terms of a liquid-gas phase transition.7 A physical interpretation of the TD unstable zone is customarily obtained by making use of the Maxwell construction. This consists in finding, for each temperature, the gas and liquid densities such that the equations µ(ρg ) = µ(ρl ) and p(ρg ) = p(ρl ) are simultaneously satisfied. The range of densities defined by ρg

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26 24

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0.3

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0.1

0.2 -3 ρ [fm ]

0.3

T [MeV]

BHF

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T [MeV]

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Fig. 2. Coexistence (circles) and spinodal (squares) regions for SNM within the BHF (left panel) and SCGF (right panel) approaches for the CDBONN interaction.

and ρl is the coexistence region, where the gas and liquid phases can coexist at constant pressure and chemical potential. Finding the corresponding spinodal and coexistence regions for each given temperature, one obtains the phase diagrams of Fig. 2 for the BHF (left panel) and the SCGF (right panel) approaches. The critical temperature for the liquid-gas phase transition of SNM corresponds to the maximum of the spinodal and coexistence lines. While for the SCGF scheme the critical temperature is Tc = 18 MeV, for BHF (with the same potential) it is substantially larger, close to Tc = 24 MeV. This large difference is basically due to the repulsive effect of holes in the energy per particle at low temperatures. That the saturation energy per particle at low temperatures and the critical temperature have to be related can be understood intuitively. At Tc the thermal effects are as large as the binding of the liquid phase and therefore one cannot form any more clusters of liquid nuclear phase. The differences in the critical density for the two approaches seem to be smaller. Finally, let us note that we do not expect that the inclusion of three body forces might change this overall picture because their effect is mainly limited to the high density region, where thermal effects are less important. Acknowledgment This work was partially supported by the NSF under Grant No. PHY-0555893 and by the MEC (Spain) and FEDER under Grant No. FIS2005-03142. References 1. 2. 3. 4. 5. 6. 7.

C. J. Pethick and G. M. Carneiro, Phys. Rev. A 7, p. 304 (1973). G. M. Carneiro and C. J. Pethick, Phys. Rev. B 11, p. 1106 (1975). T. Frick and H. M¨ uther, Phys. Rev. C 68, p. 034310 (2003). L. P. Kadanoff and G. Baym, Quantum Statistical Mechanics (Benjamin, N.Y., 1962). J. M. Luttinger and J. C. Ward, Phys. Rev. 118, p. 1417 (1960). A. Rios, A. Polls, A. Ramos and H. M¨ uther, Phys. Rev. C 74, p. 054317 (2006). J. Pochodzalla et al., Phys. Rev. Lett. 75, p. 1040 (1995).

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SMALL CLUSTERS OF para-HYDROGEN RAFAEL GUARDIOLA Departamento de Física At´ omica y Nuclear Facultad de Física, Universidad de Valencia E-46.100-Burjassot, Spain E-mail: [email protected] ´ NAVARRO JESUS IFIC (CSIC-Universidad de Valencia) Apartado Postal 22085, E-46.071-Valencia, Spain E-mail: [email protected] This paper presents a systematic study of clusters formed by parahydrogen molecules, considered as elementary particles. Among the analyzed properties there are the energy and the one-body distribution. The Many-Body method used for the analysis is the well known Diffusion Monte Carlo (DMC) with importance sampling trial functions including two- and three-body Jastrow correlations. Results are presented for the two current interactions known as Buck and Silvera. Keywords: Clusters; para-Hydrogen; magical clusters.

1. The Hydrogen Molecule Hydrogen appears in nature in the form of H2 molecules. These are the bound system of two protons and two electrons, which are well described in the framework of the familiar Born–Oppenheimer approximation. In the first stage of this approximation one assumes fixed nuclei, at a (varying) distance R, and solves the electronic problem for this configuration of nuclei. In this form one obtains an internuclear potential V (R) which in a second stage, is the input of the nuclear Schr¨ odinger equation whose eigenvalues E(ν, J) describe the spectrum of the molecule. Relevant characteristics of this internuclear potential V (R) are the equilibrium distance Re and the value at the minimum De , whose values for hydrogen are1 Re = 0.74128 ˚ A and De = 4.7466 eV. The difference between De and the molecular binding energy is the value of the lowest eigenvalue of the Schr¨ odinger equation, corresponding to zero angular momentum J = 0 and a nodeless radial function ν = 0. Very precise theoretical2 and experimental3 determinations give for the dissociation energy of hydrogen the value D = 36118.062(10)cm−1 ≈ 4.4780685eV. The molecular eigenvalues are well represented in terms of a perturbed harmonic oscillator, centered at the potential minimum, by means of Dunham expansion4

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4831.41J=3

705.54 J=2 354.57 ν=0 J=0

S=0 Fig. 1.

4273.76 J=1

S1(0)

J=2 4497.84 4161.17 ν=1 J=0

Q1(0)

-1

Excitation energy (cm )

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118.50 J=1

S=1

The low energy spectrum of hydrogen. Energies are in cm−1 .

P E(ν, J) = lm Ylm (ν + 12 )l J m (J + 1)m , known as the rovibrational formula. The most important values of these constants are5 Y10 = 4401.21, Y20 = −121.34, Y01 = 60.853, Y11 = −3.062 and Y30 = 0.813, all of them in cm−1 . There is a further consideration related to the homonuclear characteristics of hydrogen molecule, namely the spin-statistics connection. The nuclear spin may be S = 0, antisymmetric, requiring the spatial part to be symmetric, the allowed orbital angular momentum values being even. This is known as para-hydrogen. Correspondingly, one may have S = 1 and odd values of J, corresponding to ortho-hydrogen. Some molecular levels of hydrogen are drawn in Fig. 1, showing the two columns corresponding to S = 0 and S = 1. In each column few levels have been plotted, the lowest rotational levels for the quantum vibrational number ν = 0 and ν = 1. Some comments are in order: standard absorption (or spontaneous emission) of electromagnetic radiation in atoms prefers the dipolar channel, which requires |∆J = 1|, and parity change. Raman scattering, on the other hand, requires to have intermediate (virtual) states with |∆J = 1| and different parity, but being a two-photon process, final states with Jf = Ji , Ji ± 2 are reached. So, photon absorption and Raman scattering are complementary when determining the molecular spectrum. In the case of pH2 or oH2 none of the above rules is fulfilled, but the processes may go on in higher electromagnetic orders.6 Transitions in Molecular Physics are represented as Mνi →νf (Ji ), where Ji is the initial angular momentum, νi and νf are the initial and final vibrational quantum number (when νi = 0 it is suppressed) and M is a capital letter representing ∆J: Q means ∆J = 0, R means ∆J = 1, S means ∆J = 2, and so on. In Fig. 1 the vertical arrows correspond to transitions Q1 (0) = 4161.17cm−1 and S1 (0) = 4497.64cm−1 , relevant for future considerations. The relevance of this discussion about the spectrum of H2 is tied to the fact that many experiments dealing with gas, liquid or solid hydrogen, as well as with clusters , are based on the excitation of the H2 levels.

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2. Liquid and Solid Hydrogen The energy difference between oH2 and pH2 lowest levels is 170.50 K. At room temperature equilibrium hydrogen is 75% ortho and 25% para, the ratio of statistical weights. At low temperatures, near the boiling point of 20.3 K, hydrogen is nearly all in the para state. The conversion process from ortho to para is quite slow, unless a catalyst in employed to facilitate the conversion. The critical point for H2 is Tc = 33 K and PC = 1.3 MPa, making it the second coldest liquids known. It also has a very low density ρ = 0.0216 molecules per ˚ A−3 at the normal boiling point. The triple point where hydrogen begins to solidify under saturated vapor pressure is TT P = 13.8 K at PT P = 0.72MPa. Our interest in liquid and solid hydrogen lies in the fact that the infrared spectrum of H2 is affected by the presence of surrounding molecules; in other words, the intermolecular interaction modifies the intramolecular interaction. In 1955, Allin and coworkers7 measured the infrared absorption of photons by liquid and solid parahydrogen. Measurements of the Raman spectra showed a red shift of the free frequencies. A good description of the features of such phenomenon was given afterwards by van Kranendonk and Karl,5 representing the H2 spectrum in the solid in terms of few parameters describing the changes of the intramolecular potential and the coupling between the vibrational motions of nearest neighbor molecules. Experimentally, transitions like S0 (0), Q1 (0) and S1 (0), specific of parahydrogen and corresponding to excitations from the (ν = 0, J = 0) ground state to (ν = 0, J = 2), (ν = 1, J = 0) and (ν = 1, J = 2) were shifted by 1.0 ± 0.1, 8.3 ± 0.2 and 8.3 ± 0.2 cm−1 , respectively, related to two parameters. Recent measurements in solid pH2 8–10 have confirmed the mentioned shifts as well as have obtained new lines. 3. Cold p-H Gas and Dimers Interesting experiments were reported in Refs. 11–13 on absorption of infrared radiation by parahydrogen gas, at 20K, slightly above the triple point. Just focusing in the region around the Q1 (0) line there has been observed the absence of absorption at exactly the frequency of the free Q1 (0) line, and the presence of two satellites at energies above and below the free parahydrogen frequency. The interpretation of the phenomenon was that the observed lines were related to the excited states of the dimer pH2 . In practice, absorption experiments provide an indirect way of detecting the two-body ground state pH2 − pH2 and measuring its spectrum. 4. Molecular Interaction and the Dimer Hydrogen molecules interact through weak van der Waals forces that, nevertheless, are sufficiently strong to bound clusters with any number of molecules. Several forms have been derived to describe the pH2 –pH2 interaction. Two of them are of particular interest because they combine ab-initio properties with properties of

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40

He-He potential(K)

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20

0

-20

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20

30 pH-pH distance (A) o

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-40 0

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20

30

o

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He-He distance (A)

Fig. 2. Buck pH2 –pH2 interaction (left panel) and Aziz He–He interaction (rigth panel) in K plotted against the interparticle distance in ˚ A. The figures also include (dashed lines) the twobody wave functions related to these potentials.

the gas (or solid) as well as experimental information from collisions, one due to Silvera and Goldman,14 and the other of Buck et al.15 The main difference is that Silvera potential contains a repulsive long-range term (c9 /r9 ) with the objective of providing an approximation for the effective potential in a solid. In spite of the fact that m(He) ≈ 2m(pH2 ), the hydrogen interaction is so strong that the pH2 dimer has much larger binding energy (4.311K) and much shorter rms radius (5.13˚ A) than the He dimer (0.0018K and 57.33˚ A, respectively). In that sense one may say that pH2 clusters are more classical than He clusters, because the larger zero-point motion is largely compensated by the larger intermolecular attraction. When using Silvera interaction the dimer binding energy is smaller, 3.846K. The dimer has no excited states, although mathematically the state with L = 1 has binding energy of 2.034K, for Buck interaction. However, this state is forbidden by Bose statistics. Nevertheless, it may have a physical sense if one of the pH 2 molecules is excited, because then both pH2 molecules are distinguishable.5 5. Light Clusters of Parahydrogen and Quantum Many-Body Description In addition to the above mentioned indirect detection of the dimer, light clusters have been identified in a recent experiment,16 based in Q1 (0) Raman scattering, through small changes in the frequency related to the number of constituents of the cluster. The experiment clearly identifies clusters with N = 2 up to N = 8, with energy shifts between ∆ν = −0.40cm−1 for N = 2 up to ∆ν = −2.35cm−1 for N = 8. The shifts were interpreted as intermolecular effects on the intramolecular potential. Afterwards, there is a bump at N = 13, N = 33 and N = 55, which were interpreted as magical clusters . It should be mentioned that these three values are actually extrapolations from smaller clusters and presumably are approximate. During the years 1992–1994 there was some interest in studying parahydrogen clusters, using the Path Integral Monte Carlo (PIMC) method17,18 or Diffusion

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Monte Carlo (DMC) method.19–21 These studies were devoted to clusters with scattered number of molecules (7, 13, 19, 33), presumably motivated by the geometric shapes found in classical Lennard–Jones clusters, and also with the aim of testing some form of superfluidity. Afterwards the interest was shifted to parahydrogen doped clusters,22–27 with special focus in cold spectroscopy and the role of superfluidity. These studies were basically concerned with very light clusters. Finally in the last six months several papers have appeared with a systematic study of clusters from N = 3 to N = 40 molecules with the main objective of checking the magical character found in the experiment of Tejeda et al.16 The Many Body techniques used are DMC,28 a Path Integral method adapted to zero temperature (PIGS)29 and PIMC30–32 and using Buck potential,28 Silvera potential30–32 or both.29 Whereas results up to N = 20 are in agreement for all mentioned calculations, there are noticeably differences for larger values, particularly for N > 30. For this reason we have improved our previous analysis28 in order to get very precise results within our computational capacities. 5.1. Variational trial function and DMC algorithm DMC is based in a short-time approximation of the Green’s function related to the imaginary time Schr¨ odinger equation. In this way, an initial wave function ΦT (t = 0) evolves to the exact ground state wave function Ψgs at large t after many shorttime steps τ . We have used the O(τ 3 ) approximation as described in Refs. 33,34, which provides energies O(τ 2 ). The bare DMC procedure is significantly improved when using a good importance sampling wave function, the main effect being the reduction of the variance of the stochastic procedure. We have used a Jastrow wave function with two- and three-body correlations: ΦT = exp(¯ u2 + u3 ), with u ¯2 =

X 2 u2 (rij ) + λT ξ 2 (rij )rij

,

i<j

where

# " X p5 p1 u2 (r) = − 5 + r rij ij i<j

,

G` =

X i6=`

ξ(rli )rli

u3 = −

,

λT X G` G` 2 `

(r − sT )2 ξ(r) = exp − wT2

which includes three-body correlations35 but still having O(N 2 ) computational complexity. The form ΦT = exp(u2 ) is the standard two-body Jastrow wave function. The time step adjustment is described in Table 1: from calculations at the relative large steps 0.001 and 0.005K−1, we obtain the Richardson extrapolated value, based on the τ expansion E(τ ) = E(0) + Cτ 2 + · · · , which turns out to be very close to the calculations with much smaller time steps. However, the statistical error

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remains still high. This table checks that the algorithm has been properly coded and suggests using τ = 0.0001K−1 for the massive calculations with a minimal bias. Table 1. Determination of optimal time step τ from different evaluations of the binding energy (B) of several clusters. The row labelled Extrapolation is obtained from the previous two rows by the combination (4B(N, 0.0005) − B(N, 0.001))/3. Energies and statistical errors are in K. τ

B(10)

Error

B(20)

Err

B(30)

Err

0.001 0.0005 Extrapolation

183.47 185.91 186.72

0.05 0.06 0.09

559.28 566.56 568.99

0.17 0.17 0.28

1006.4 1020.0 1024.5

0.3 0.4 0.5

0.0001 0.00002

186.93 186.72

0.06 0.03

569.16 569.48

0.12 0.07

1025.2 1024.8

0.2 0.1

A further improvement of the calculation regards the estimate of the statistical error. Because of the stochastic nature of Monte Carlo algorithms, successive samples are strongly correlated, and the typical way of estimating the variance, hH 2 i − hHi2 , may result in too optimistic values for the statistical error. To avoid these correlations one may use a block average procedure or, as we have done here, one may compute a number of times the binding energy, with independent and randomized runs, and estimate the variance with these results. This requires a formidable increasing in computational time, but the obtained standard deviations are very precisely computed. 5.2. Binding energies DMC results are compared with variational Monte Carlo (VMC) results in Fig. 3, where the binding energy per particle is plotted as a function of the number of constituents. It is interesting to note the effect of three-body correlations, which provides half of the missing energy of the standard pair-correlated Jastrow wave function. The figure also shows the separate contribution of potential and kinetic energies for m = 2 amu (parahydrogen), m = 4 amu (orthodeuterium) and the classical potential energy, to illustrate the transition from quantum to classical systems. In Fig. 4 we have plotted the DMC dissociation energies µ = B(N ) − B(N − 1). The left panel shows our DMC results for N = 3 to 40, both for Buck and Silvera potentials. The results of Buck potential supersede those of Ref. 28 as they have been obtained with both three-body correlations and an accurate statistical analysis of the error. The calculations for the Silvera potential also include triplet correlations, but the statistics is not as good as in the previous case, as these have larger error bars. It should be mentioned that the values of the dissociation energies are very sensitive to small errors, specially for large clusters, which should be computed with

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Variational triplet Variational Jastrow

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500 0 -500 -1000

2 amu

-1500

4 amu

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Classical

-2500 0

0 30

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Fig. 3. On the left panel the energies per molecule (K) are plotted as a function of the number of molecules of the cluster obtained with DMC (black squares), VMC with triplet correlations (circles) and VMC with pair correlations (open squares). On the right panel the potential and kinetic energies are plotted for pH2 and oD2 , as well as the classical potential energy. Both figures correspond to Buck potential.

60

55

Dissociation energy (K)

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Dissociation energy (K)

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50

45

40

10 0 0

5

10

15

20

25

30

Number of molecules

35

40

35 20

24

28

32

36

40

Number of molecules

Fig. 4. The figure on the left represents the dissociation energy (in K) obtained with DMC for Buck (black squares) and Silvera (open circles) potentials. The right panel compares our DMC results with PIMC calculations32 (open squares, Silvera potential).

a precision of a fraction of a Kelvin (one part in 10000) in order to have a precision of 1% for µ. Up to N ≈ 25, our DMC results are indistinguishable from the the PIMC ones of Ref. 32 at T = 0.5K or those of Ref. 31 at T = 1 K. However, differences appear beyond this size. On the right panel of Fig. 4 we have plotted in an amplified scale DMC and PIMC results. The main physical result of this picture is the presence of a neat peak at N = 13

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in the dissociation energy, indicating the magical character of this cluster. This peak is clearly present for both interactions. Beyond this point, the only possible anomaly appears at N = 36 for Buck potential, but µ(36) is only 1.5 K higher than the neighbors, in other words, is not too magical. In the case of Silvera potential our calculations have not as good a statistics, and beyond N = 25 the values of µ are scattered around a continuous line, and within the error bars no anomalies are present. This is in clear contrast with PIMC calculations, which show very prominent peaks at N = 26, 29, 32, 34 and 39. To explain this disagreement two possibilities can be mentioned: • The importance sampling function of DMC puts a strong constraint on the stochastic random walk, thus producing biased results. • The assumed error of PIMC calculations is too small, and the observed peaks are just statistical fluctuations. Our present study already discards another possibility, namely the difference between Buck and Silvera potentials as being the origin of the disagreement. In fact, we observe that even if the energies are different (more binding energy for Buck potential) there are no qualitative differences. 5.3. The shape of para-Hydrogen clusters

e) ts en itu st on fC ro be um N

N

um

be

ro

fC

on

st

itu

en

ts

(p

(H

H

)

Figure 5 compares the one-body densities of para-Hydrogen and 4 He clusters, norR malized to the number of particles, ρ(r)dr = N . One observes a clear shell structure in para-Hydrogen, which is only insinuated in 4 He. Para-Hydrogen densities are compatible with a solid-like structure of the type of classical Lennard–Jones crystals but the distributions do not serve to establish such a structure.

-3

0.04 40

o

o

50

Density distribution (A )

40

-3

Density distribution (A )

50 0.20

0.15

30

0.10

20

0.05

10

0.00 0

2

4

6

8o

Distance to Center-of-Mass (A)

10

0

0.03 30

0.02

20

0.01

0.00 0

10

2

4

6

8

10

0

o

Distance to Center-of-Mass (A)

Fig. 5. One body densities of para-Hydrogen (left) and 4 He (right) clusters. Note that vertical scales are different.

To interpret such distributions we have found convenient to fit functions with

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overlapping Gaussians ρ(r) ≈

X k

(r − ck )2 Ak exp − 2σk2

,

where the number of Gaussians depends on the considered cluster. The quantities ck and σk represent the centroids and widths of the shells, respectively. By integrating separately each Gaussian we may determine how many particles lie in each shell. This is shown in Fig. 6. 10

Number of particles in a shell

40

o

Centroids and widths (A)

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6

4

2 0

30

20

10

0 0

10

20

30


40

50

0

10

20

30

40

50

Number of Constituents

Fig. 6. Gaussian fits to the one-body density. On the left we represent the centroids σ k . The error bars represent the widths σk . On the right the number of particles assigned to each shell, rounded to the nearest integer, are represented.

In spite of the very strong peak appearing at the origin, our analysis shows that there is a single particle there. The filling of shells is unusual, in the sense that one might expect an ordered filling scheme, while it turns out that the occupancies of two shells grows simultaneously. Only after N = 30 there seems to have a more traditional pattern: one particle at the origin, 13 particles (a very odd number) in the first shell and a growing number of particles in the most external shell. However, calculations of heavier clusters show that around N=70 the particle near the origin disappears. The shell structure may be related to the icosahedral symmetry of classical Lennard–Jones clusters, in the same form that magical numbers have some parallelism with classical descriptions. 6. Conclusion Hydrogen clusters are fascinating, with a richness of properties not found in the more familiar 4 He clusters. There are still some open problems: PIMC calculations should be revisited, and other variational forms for DMC should be experimented. Finally, one should fill the gap between T=0 and non null temperatures by studying the excitation spectrum of clusters. Acknowledgments The authors acknowledge stimulating conversations with Jan Peter Toennies and Salvador Montero. This work is supported by grants FIS2004-00912

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(MCyT/FEDER, Spain), and ACOMP07-003 (Generalitat Valenciana, Spain). References 1. W. Kolos and C.C.J. Roothaan, Rev. Mod. Phys. 32, 219 (1960). 2. L. Wolniewicz, J. Chem. Phys. 103, 1792 (1995). 3. Y.P. Zhang, C.H. Cheng, J.T. Kim, J. Stanojevic, and E.E. Eyler, Phys. Rev. Lett. 92, 203003 (2004). 4. J.L. Dunham, Phys. Rev. 41, 721 (1932). 5. J. Van Kranendonk and G. Karl, Rev. Mod. Phys. 40, 531 (1968). 6. L. Wolniewicz, I. Simbotin and A. Dalgarno, Astrophys. J. Suppl. Ser. 115 293 (1998). 7. E. J. Allin, W.F. Hare, and R.E. MacDonald, Phys. Rev. 98, 554 (1955). 8. W. Ivancic, T. K. Balasubramanian, J.R. Gaines, and K. Narahari Rao, J. Chem. Phys. 74, 1508 (1981). 9. M. Okumura, M. Chan, and T. Oka, Phys. Rev. Lett. 62, 32 (1989). 10. A. Nucara, P. Calvani, and B. Ruzicka, Phys. Rev. B 49, 6672 (1994). 11. A. Watanabe and H.L. Welsh, Phys. Rev. Lett. 13, 810 (1964). 12. A.R.W. McKellar, J. Chem. Phys. 92, 3261 (1990). 13. A.R.W. McKellar, J. Chem. Phys. 95, 3081 (1991). 14. I.F. Silvera, V.V. Goldman, J. Chem. Phys. 69, 4209 (1978). 15. U. Buck, F. Huisken, A. Kohlhase, D. Otten, and J. Schaeffer, J. Chem. Phys. 78, 4439 (1983). 16. G. Tejeda, J.M. Fern´ andez, S. Montero, D. Blume, and J. P. Toennies, Phys. Rev. Lett. 92, 223401 (2004). 17. P. Sindzingre, D.M.Ceperley and M.L.Klein, Phys. Rev. Lett. 67, 14 (1992). 18. D. Scharf, M.L. Klein and G.J. Martyna, J. Chem. Phys. 97, 3590 (1992). 19. M.A. McMahon, R.N. Barnett, and K.B. Whaley, J. Chem. Phys. 99, 8816 (1993). 20. M.A. McMahon, K.B. Whaley, Chem. Phys. 182, 119 (1994). 21. E. Cheng, M.A. McMahon, and K.B. Whaley, J. Chem. Phys. 104, 2669 (1996). 22. Yongkyung Kwon and K.B. Whaley, Phys. Rev. Lett. 89, 273401 (2002). 23. F. Paesani, R. E. Zillich, and K. B. Whaley, J. Chem. Phys. 119, 11682 (2003). 24. J. Tang and A. R. W. McKellar, J. Chem. Phys. 121, 3088 (2004). 25. C. H. Mak, S. Zakharov, and D. B. Spry, J. Chem. Phys. 122, 104301 (2005). 26. S. Baroni and S. Moroni, ChemPhysChem 6, 1884 (2005). 27. S. Moroni, M. Botti, S. De Palo and A. R. W. McKellar, J. Chem. Phys. 122, 094314 (2005). 28. R. Guardiola and J. Navarro, Phys. Rev. A 74, 025201 (2006). There is a misprint in Table I of this work. The entry for B/N at N = 23 should read 30.43(2), instead of the quoted 29.94(2). We acknowledge J.E. Cuervo for having detected the erratum. See also Ref. 31. 29. J.E. Cuervo and P.N. Roy, J. Chem. Phys. 125, 124314 (2006). 30. F. Mezzacapo and M. Boninsegni, Phys. Rev. Lett. 97, 045301 (2006). 31. F. Mezzacapo and M. Boninsegni, Phys. Rev. A 75, 033201 (2007). 32. S. A. Khairallah, M. B. Sevryuk, D.M.Ceperley and J. P. Toennies, Phys. Rev. Lett. 98, 183401 (2007). 33. J. Vrbik and S. M. Rothstein, J. Comput. Phys. 63, 130 (1986). 34. S. A. Chin, Phys. Rev. A 42, 6991 (1990). 35. K.E. Schmidt, M.A.Lee, M.H. Kalos and G.V. Chester, Phys. Rev. Lett. 47, 807 (1981).

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ADHESIVE FORCES ON HELIUM IN NONTRIVIAL GEOMETRIES ´ E. S. HERNANDEZ Departmento de F´ısica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina ∗ E-mail: [email protected] A. HERNANDO, R. MAYOL and M. PI Departament d’Estructura i Constituents de la Mat` eria IN2 UB, Facultat de F´ısica, Universitat de Barcelona, 08028 Barcelona, Spain We have recently shown that the adsorption potential for a solid with a planar surface can be derived from a unique elementary source field which can be applied to other geometries. We review this procedure, discuss limitations of previous approximations and present new results for adsorption of helium in hexagonal pores. Keywords: Adsorption potentials; arbitrary shapes; helium.

1. Introduction When a sample of solid matter is exposed to a vapor at fixed temperature and pressure, condensation on the surfaces occurs if the thermodynamic grand potential Ω = E − µN , where E, µ and N respectively denote the total energy, chemical potential and number of atoms, is nonpositive. The strength and shape of the adsorption field U (r) experienced by an atom in the vapor is a fundamental ingredient for the determination of the transition which, in principle, can be expressed as a summation over particles in the substrate U (r) =

X i

V (r − ri ) ≈

Z

dr0 n(r0 ) V (r − r0 )

(1)

where the last integral is an approximation that assumes a continuous matter distribution with local density n(r). Models for the pair interaction V (r) permit calculations of the external potential U in simple geometries such as planes, cylinders or spheres. However, for surfaces presenting angles like wedges, polygonal pores, and patterned substrates, the standard approximation reported in the literature1–4 may overestimate the total strength. In a recent work,5 we have shown that if the adsorption potential for a solid with a planar surface is known, there exists a unique source or ”pair potential” which can be later integrated according to Eq. (1) for

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any geometry, restricted by numerical feasibility. We summarize this procedure in Sec. II, and in Sec. III we compare with previous approximations and discuss various aspects of adsorption of helium in hexagonal pores. 2. The Elementary Potential One of the most popular choices for the pair potential in Eq. (1) is the Lenard-Jones (LJ) potential VLJ (r), whose integration over a semi-infinite solid of bulk density ρ0 with a planar surface at z = 0 gives the well-known field U (z) ≡ V3−9 (z). The LJ potential —generally speaking, any power-law potential r −n — is integrable in many geometries with high symmetry like cylindrical and spherical pores or cavities, sheets and wedges. For metallic planar half-solids, an accurate adsorption field U (z) ≡ VCCZ (z) has been derived by Chizmeshya, Cole and Zaremba;6 being ab-initio, the CCZ field cannot be trivially expressed through Eq. (1) in terms of a pair potential Vpair (r). In Ref. 5, we have shown that given a reference adhesive field Uref (z), is it possible to derive an elementary potential V (r) such that Uref (z) = ρ0

Z

0

dz 0 −∞

Z Z

dx0 dy 0 V (r − r0 ).

(2)

The above integral equation for the source field is exactly solvable with 00

V (z) =

Uref (z) 2πρ0 z

(3)

which can be verified for the LJ family where this simple algebraic relation links a planar V3−9 (z) with its source VLJ . For other matter distributions n(r), the adsorption field can be then computed with Eq. (1). Tests of numerical accuracy and a detailed discussion of several solutions and applications have been presented in Ref. 5. 3. Hexagonal Pores As discussed in Ref. 2, FSM-16 is a silica material whose adsorption strength on helium is not accurately known, so that at least two different V3−9 sets of parameters, respectively identified as A and B, have been employed. We have verified that for an FSM-16 hexagonal pore, the procedures (i) summing the contributions from six planar substrates2 and (ii) integrating the original LJ, give sizeable discrepancies of near 40% at the minimum for both fields A and B, indicating a large overcounting near the vertices. These effects for a pore with side of 14 ˚ A can be viewed in Fig. 1, whose lower panel shows the energy per particle E/N and chemical potential µ of helium particles as functions of their linear density N/L, obtained by a finite range density functional calculation in the manner of Refs. 3,4 and with both procedures (i) and (ii) to compute the external field for parametrization B. The grand potential

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per particle Ω/N is shown in the upper panel, and the arrows point at the condensation transition where helium starts filling the hexagon corners in a stable fashion.

2 Ω / N (K)

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Six planar V3-9 potentials Integrated VLJ

-6 -110 -120 -130

E / N (K) µ (K)

-140 -150 0

0.2

0.4 0.6 -1 N / L (Å )

0.8

1

Fig. 1. Upper panel: the grand potential per particle for helium atoms as a function of the linear density. Lower panel: energy per particle (thick lines) and chemical potential (thin lines).

Table 1. SUBSTRATE FSM-16, A FSM-16, B Mg Li Na K Rb Cs

Ground state energies. εsum −εintegral εsum εintegral ε integral

−178.36 −140.77 −51.04 −23.98 −16.35 −10.41 −9.45 −9.01

−135.98 −105.36 −37.62 −17.22 −11.58 −7.24 −6.54 −6.22

0.31 0.34 0.36 0.39 0.41 0.44 0.44 0.45

The ground-state energy of a single particle in the pore, corresponding to the zero-density ordinate in the lower panel of Fig. 1, is overestimated by more than 30%

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0 0.02 -5 0.01

-10 -10 -5

0 5 x (Å)

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0 0

integration summation 10

5 d (Å)

Fig. 2. Left panel: contour lines for helium density in an hexagonal Cs pore at N/L = 7 ˚ A−1 . Right panel: density along the x– and z– axis, d being the distance from the center of the pore.

by the summation of six planes. In Table 1 we display the ground-state energy for a 4 He atom inside a FSM-16 pore with potentials A and B, together with those for metallic pores with CCZ potentials. The relative deviation in the righmost column increases as the overall adsorption strength and corresponding ground-state energy become smaller. Adsorption isotherms for metallic pores yield results very similar to those in Fig. 1. A typical helium density landscape for a Cs hexagonal pore with a linear density of 7 ˚ A−1 located immediately after the condensation threshold, as computed with CCZ potentials, is plotted in the left panel of Fig. 2. While these contour plots are essentially insensitive to the criterion adopted for the construction of the hexagon potential, the corresponding condensation transitions, which take place at very close linear densities, demand noticeably different chemical potentials, as in the case depicted in Fig. 1 and with the change of scale that follows from Table 1. The right panel in this figure shows the helium density along the diagonal (x-axis) and the height (z-axis) of the pore. The high structure is clearly visible in the multiple oscillations of the density in both directions. It is also apparent that for this weak adsorber the distinct depths of the external field do not yield substantial differences in the size of the density peaks, at variance with the case of the strong FSM-16 fields where such differences are visible. References 1. M.W. Cole, F. Ancilotto and S. M. Gatica, J. Low Temp. Phys. 138, 195 (2005). 2. M. Rossi, D. E. Galli and L. Reatto, Phys. Rev. B 72, 064516 (2005). 3. E. S. Hern´ andez, F. Ancilotto, M. Barranco, R. Mayol and M. Pi, Phys. Rev. B 73, 245406 (2006). 4. R. Mayol, F. Ancilotto, M. Barranco, E. S. Hern´ andez and M. Pi, J. Low Temp. Phys. 148 August 2007 (online version available). 5. A. Hernando, E. S. Hern´ andez, R. Mayol and M. Pi, to be published. 6. A. Chizmeshya, M. W. Cole, and E. Zaremba, J. Low Temp. Phys. 110, 677 (1998).

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ROTATIONAL SPECTRA IN HELIUM-4 CLUSTERS AND DROPLETS: SIZE DEPENDENCE AND ROTATIONAL LINEWIDTH ROBERT E. ZILLICH Institut f¨ ur Theoretische Physik, Johannes-Kepler Universit¨ at, Altenbergerstr. 69, 4040 Linz, Austria E-mail: [email protected] K. BIRGITTA WHALEY Pitzer Center, Department of Chemistry, Berkeley, CA 94720, USA E-mail: [email protected] We have implemented the correlated basis function method for molecule dynamics in 4 He using ground state properties obtained by unbiased diffusion Monte Carlo simulation. We have calculated rotational spectra of linear molecules solvated in superfluid 4 He. We present a simple model for solvation in finite 4 He droplets, that is able to explain the observed Lorentzian lineshape of CO rotation spectra by inhomogeneous broadening originating from the wide size distribution of experimentally generated droplets. Keywords: Helium; correlated basis functions; matrix spectroscopy.

1. Introduction Advances in experimental techniques in helium droplet spectroscopy allow investigation of the 4 He cluster size dependence of rotational spectra of solvated molecules up to large cluster sizes on the order of N ≈ 100. Hence recent experiments1,2 bridge the gap between earlier studies on small clusters, which showed the onset of superfluidity, and spectra obtained in large 4 He droplets containing thousands of atoms. In the range N < 100, these experiments suggest a slow, oscillatory convergence to the large droplet limit. We have found a size-effect for CO, where reptation and path integral MC simulations3,4 of small clusters of N = O(102 ) gave a significantly higher value for the rotational constant than the calculated rotational spectrum of CO in bulk helium.5 We attributed this to the coupling to long wave length collective modes that are present only in large 4 He droplets, but not in small clusters. The same type of coupling leads to a large rotational linewidth in bulk 4 He. We first review correlated basis function theory (CBF) applied, in conjunction with diffusion Monte Carlo (DMC) simulations, to chromophore spectra in 4 He. We then present a simple

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model for medium-sized to large droplets (1000 < N < ∞) which explains the observation5 of the Lorentzian line shape of the R(0) line in terms of inhomogeneous broadening due to the experimental droplet size distribution. The input in our calculations is the Hamiltonian H of a linear molecule and N 4 He atoms,6 characterized by the molecule and 4 He mass, gas phase rotational constant B, and the molecule-4 He and 4 He-4 He potential energy surfaces. The goal is to reproduce the rotational spectrum SJ (ω) for the 0 → J transition of the molecule solvated in 4 He without any further input than H. 2. CBF-DMC Approach for CO Assuming we know the ground state Φ0 of H, we write the time-dependent wave function in the form |Φ(t)i = eδU (t) |Φ0 i/hΦ0 |e<eδU (t) |Φ0 i and determine δU (t) using stationarity of the action integral Z ∂ L = dt hΦ(t)|H + Uext − i~ |Φ(t)i = Min. (1) ∂t In the case of a general correlation function δU (t), Eq. (1) is equivalent to the time-dependent Schr¨ odinger equation. For a practical calculation of the solution of Eq. (1) we approximate δU (t) by neglecting correlations higher than pair correP lations: δU = δu1 (r0 , Ω) + N i=1 δu2 (r0 , ri , Ω). Here r0 and Ω are coordinate and orientation, respectively, of the molecule, and {ri } are the 4 He coordinates. Then Eq. (1) becomes a coupled pair of Euler–Lagrange equations δL = 0, δu1 (r0 , Ω)

δL =0 δu2 (r0 , r1 , Ω)

which we solve using linear response theory since the perturbation Uext (applied laser field) is weak. The rotational energies are “renormalized” by a self energy, EJ = BJ(J + 1) + ΣJ (EJ ). The derivation of ΣJ can be found in Ref. 6. The corresponding rotational spectrum as a function of the ilaser frequency ω is given by h i−1 (2) S(ω) = =m ~ω − BJ(J + 1) − ΣJ (~ω)

which is a spectrum of Lorentzian peaks at EJ of width =mΣ(EJ ) if =mΣ(EJ ) is small. In the linear response approach, the self energy ΣJ is a functional of ground state quantities, in particular of the pair distribution g(r, cos θ) between molecule and He atoms. How to obtain g(r, cos θ)? A many-body approach which would be consistent with the formualated CBF approach is the hypernetted-chain/Euler-Langrange method (HNC/EL). The implementation of HNC/EL for anisotropic impurities in 4 He is feasible, but would be more involved than the CBF method for excited states because of the nonlinearity of the HNC/EL equations. Another option is to use quantum Monte Carlo simulations which are ideally suited to provide exact ground

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297 Table 1. Comparison of experimental and CBF-DMC ratios Beff /B. molecule

experiment

CBF-DMC

HCN HCCH CO

0.814 0.89 0.63

0.85 0.91 0.67

state properties. We choose diffusion Monte Carlo (DMC) to obtain g(r, cos θ) by unbiased sampling (using the descendent weighting method7 ). The rotational energies EJ and the spectrum SJ (ω) have been calculated for HCN/DCN6 and HCCH,8 that were shown to be in good agreement with experimental results in large 4 He droplets, typically slightly higher than the experimental value (Table 1). For CO, calculations of EJ for J = 1 gave a surprisingly large reduction of 69% — recently improved to 67% by increasing the DMC population size — with respect to the gas phase value 2B (neglecting centrifugal distortion which is small in the gas phase). But again they were in good agreement with experiment (63%), including the effect on the spectrum upon isotopic substitution.5 The reduction was significantly larger than in small clusters N < 100.3,4,9 The CBF calculation showed that the larger reduction of EJ as compared to other molecules in a similar range of mass and B can be attributed to the large dipole component of g(r, cos θ). An interesting feature of the experimental rovibrational spectra was the Lorentzian shape of the J = 0 → 1 (R(0)) line. In Ref. 5 we attributed this to homogeneous broadening due to coupling of phonons and particle-in-the-box states of the molecule. In the following we present an alternative, albeit quantitatively indistinguishable, explanation of the observed Lorentzian lineshape. Finite droplet size affects the rotational dynamics of a molecule in several ways: the elementary excitations of 4 He, the phonon-roton spectrum, become discrete; in addition to that, surface modes (ripplons) are present; the molecule is subject to an effective confinement potential keeping it inside the droplet. The first effect can be accounted for by a simplistic model based on the CBF result in bulk 4 He. The phonon-roton spectrum is discretized by the condition that the corresponding phonon-roton modes in the Feynman approximation (i.e. Bessel functions) have a node at the droplet radius R (details of this minimum model will be given elsewhere). 3 Figure 1(a) shows the spectrum SN (ω) as function of N = ρ0 4π 3 R (ρ0 = −3 0.02186˚ A ). In order to give an impression of the line strength we have slightly broadened SN (ω) (which for given N consists of delta functions) by a small imaginary part in the denominator of Eq. (2). We observe a series of well-separated resonances. Experimental generation of 4 He droplets via nozzle expansion produces a wide distribution of sizes, often assumed to be a log-normal distribution P (N ).10 P We therefore average the spectrum S(ω) = N P (N )SN (ω) in order to compare with experiment. This is shown in Fig. 1(b) for a number of “width parameters” ¯ is well reproduced. We note that d.10 The dependence on average droplets size N

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line width [K]

3.96

4

3.92

3

d=0.5 d=0.6 d=0.7 d=0.9

0.018

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104

105

106

3.88

0.016

0.030 0.028

0.014

0.026

0.012

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0.022

3.888

peak position [K]

E [K]

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3.84 3.80

3.472

d=0.5 d=0.6 d=0.7 d=0.9

3.884

3.468

3.880 3.464 3.876

0

5000

10000

15000 N

20000

3.460

25000 1000

3000

5000

7000

9000

N

(a)

11000

1000

3000 N

(b)

Fig. 1. (a) The rotational spectrum S1 (ω) as function of droplet size N ; (b) Peak position and ¯ , CBF-DMC result (left) and experiment (right). linewidth as function of N

P (N ) typically ranges over no more than one or two resonances (shown in Fig. 1(a)). Hence we are still far from the bulk limit of homogeneous broadening. Nevertheless, the inhomogeneous broadening due to the size distribution leads to an almost per¯ as small as 2000, which is in qualitative fect Lorentzian lineshape (not shown) for N agreement with the experimental observation.5 3. Conclusion We found that the inhomogeneous spectrum of a single particle excitation (molecule rotation) coupled to an isolated collective excitation (phonon) of a small system (4 He droplet) is Lorentzian provided that there is a sufficient statistical spread of system sizes to cover a full resonance. The spectrum is the same as the homogeneous spectrum of this single particle excitation in the bulk limit, although the average size is still very far away from this limit. Our result is consistent with a model based on the coupling of a bright state to a discrete bath of harmonic dark states that are subject to a size-dependent energy shift.11 References 1. Y. Xu, W. Jager, J. Tang and A. R. W. McKellar, Phys. Rev. Lett. 91, p. 163401 (2003). 2. A. R. W. McKellar, Y. Xu and W. J¨ ager, Phys. Rev. Lett. 97, p. 183401 (2006). 3. P. Cazzato, S. Paolini, S. Moroni and S. Baroni, J. Chem. Phys. 120, p. 9071 (2004). 4. R. E. Zillich, F. Paesani, Y. Kwon and K. B. Whaley, J. Chem. Phys. 123, p. 114301 (2005). 5. K. van Haeften, S. Rudolph, I. Simanovski, M. Havenith, R. E. Zillich and K. B. Whaley, Phys. Rev. B 73, p. 054502 (2006). 6. R. Zillich and K. B. Whaley, Phys. Rev. B 6, p. 104517 (2004). 7. J. Casulleras and J. Boronat, Phys. Rev. B 52, p. 3654 (1995). 8. R. Zillich, Y. Kwon and K. B. Whaley, Phys. Rev. Lett. 93, p. 250401 (2004). 9. J. Tang and A. R. W. McKellar, J. Chem. Phys. 119, p. 754 (2004).

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10. B. Dick and A. Slenczka, J. Chem. Phys. 115, p. 10206 (2001). 11. K. K. Lehmann, J. Chem. Phys. 126, p. 024108 (2007).

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MICROSCOPIC STUDIES OF SOLID 4 HE WITH PATH INTEGRAL PROJECTOR MONTE CARLO M. ROSSI, R. ROTA, E. VITALI, D.E. GALLI∗ and L. REATTO Dipartimento di Fisica, Universit` a degli Studi di Milano, via Celoria 16, 20133 Milano, Italy ∗ E-mail: [email protected] We have investigated the ground state properties of solid 4 He with the Shadow Path Integral Ground State method. This exact T = 0 K projector method allows to describes quantum solids without introducing any a priori equilibrium position. We have found that the efficiency in computing off-diagonal properties in the solid phase sensibly improves when the direct sampling of permutations, in principle not required, is introduced. We have computed the exact one-body density matrix (OBDM) in large commensurate 4 He crystal finding a decreasing condensate fraction with increasing imaginary time of projection, making our result not conclusive on the presence of Bose–Einstein condensation in bulk solid 4 He. We can only give an upper bound of 2.5 × 10−8 on the condensate fraction. We have exploited the Shadow Path Integral Ground State (SPIGS) method to study also 4 He crystal containing grain boundaries by computing the related surface energy and the OBDM along these defects. We have found that also highly symmetrical grain boundaries have a finite condensate fraction. We have also derived a route for the estimation of the true equilibrium concentration of vacancies xv in bulk T = 0 k solid 4 He, which is shown to be finite, x = (1.4 ± 0.1) × 10−3 at the melting density, when v computed with the variational shadow wave function technique. Keywords: Supersolid; path integral; projector quantum Monte Carlo.

1. Introduction The low-temperature physics of 4 He samples has been a continuous test ground of many-body theories; though displaying all the features of a strongly interacting system, a collection of 4 He atoms has a very simple effective Hamiltonian: a model of spinless and structureless bosons interacting through a two-body interaction potential has been proved to provide a very accurate description of the properties of liquid and solid Helium, so that, when dealing with 4 He, one does not have to face the added complexities related to Fermi statistics or to nuclear Physics. The phenomenon of superfluidity in liquid Helium, displaying in a macroscopic domain the quantum laws of nature related to the boson indistinguishability of the atoms, has been the object of several decades of theoretical efforts; the development of very accurate computer simulations methods has provided the possibility to put light into the physical mechanisms which govern such striking experimental observations. It is known that, under pressure, at very low temperature, a sample of Helium

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atoms solidifies. For many years it has been argued that quantum coherence phenomena, such as Bose Einstein Condensation (BEC) and superfluidity, could take place even in the crystalline phase if the compromise between particles localization induced by the crystalline order and particles delocalization due to the zero-point motion of the atoms allowed the boson nature of the particles to play an important role giving rise to a supersolid phase. The supersolid state of matter, formerly appeared only in the speculations of some eminent theoreticians,1–3 has been elusive for decades4 up to its first probable evidence in torsional oscillator experiments with solid 4 He.5 Now, after three years of renewed interest and intense experimental and theoretical studies, physicists still find themselves in front of a puzzle of signs difficult to recompose.6 Non classical rotational inertia (NCRI) effects in torsional oscillator experiments with solid 4 He (which have been observed also in confined samples7 ) have been observed in a number of laboratories8–11 and recently also the robustness of these effects has been tested by working with single crystals.12 The whole experimental scenario reached after three years of research activity is not capable to clearly discern whether such effects arise from the true thermal equilibrium state of the system or whether they are due to non equilibrium properties. From one side, if one omits the amazing direct observation of superfluid effects in presence of grain boundaries13 and the specific heat peak detected near 80 mK in solid 4 He,12 NCRI effects in torsional oscillator are still the only manifestation of the elusive supersolid phase; from the other side it is clear that such effects are strongly affected by disorder but what is not clear is to what extent the disorder plays a role, because there are conflicting conclusions on the effects of annealing processes and now there is also evidence of a variable amount of NCRI in different samples even in single crystals after annealing.12 Very accurate “ab-initio” microscopic simulation methods, whose importance in exploring the liquid phase has been outstanding, could give a deep insight into solid Helium physics. We have used the SPIGS method14 to investigate the zerotemperature properties of solid 4 He. In the first section we introduce the SPIGS technique; the evaluation of the one body density matrix in both a commensurate and incommensurate sample of solid Helium is reported in the second section. Then we deal with defects, vacancies and grain-boundaries: we describe a method to evaluate the concentration of zero-point vacancies, which are known to help the delocalization of the atoms thus favoring BEC in the system, and to study their dynamics, and, in the following section, we report our diagonal and off-diagonal results in presence of grain-boundaries.

2. Path Integral Projector Monte Carlo Methods The Path Integral Ground State Method (PIGS) is a T = 0 K projector Quantum Monte Carlo method which allows the calculation of “exact” ground state averages in a quantum system using a wave function Ψτ that asymptotically approaches the unknown ground state Ψ0 .15 Ψτ is obtained via iterative applications of the

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operator e−δH (this action will be called projection in the following) on an initial trial state, ΨT , characterized by a non-zero overlap with the true ground state ˆ ˆ ˆ is the Hamiltonian operator of the Ψ0 : Ψτ = (e−δH )P ΨT = e−τ H ΨT , where H quantum system and δ = τ /P . The projection procedure exponentially removes (as a function of τ ) from Ψτ any overlap of ΨT with the excited states. In the position representation the corresponding wave function Ψτ (R) = hR|Ψτ i is expressed as a succession of P convolutions with the imaginary time propagator G(R, R 0 , δ) = ˆ hR|e−δH |R0 i to give the Path Integral approximation of the ground state wave function: Z Y P [dRj G(Rj+1 , Rj , δ)] ΨT (R1 ) (1) Ψτ (R) = j=1

where RP +1 ≡ R, ΨT (R) = hR|ΨT i and generally R = {~r1 , ~r2 , ..., ~rN } represents a set of coordinates for the N particles in the quantum system. If δ is chosen sufficiently small, accurate approximations for G(R, R 0 , δ) are known even for a strongly interacting system like a collection of 4 He atoms in the solid phase.16 When only one projection step is considered and the imaginary time propagator is approximated via a variationally optimized “primitive” density matrix,16 the well known variational shadow wave function (SWF) technique is recovered.17,18 With SWF, which is translationally invariant and contains implicitly correlations at all the orders in the number of particles, the solid phase emerges as a result of a spontaneously broken translational symmetry.17 This is done in a so efficient way that it is possible to describe the solid and the liquid phase with the same functional form since there is no need of a priori known equilibrium positions for the crystalline phase. Moreover SWF presently represents the best variational description of liquid and solid Helium18 in the sense that it gives the lowest energy in both the phases. The Shadow Path Integral Ground State method (SPIGS) corresponds to a PIGS method in which the projection procedure is applied to a SWF.14 Due to the fact that the first imaginary time projection step is done with the SWF technique, SPIGS method allows to describe the solid phase without explicitly breaking the translational symmetry. To our knowledge SPIGS is the only projector quantum Monte Carlo method which provides the possibility to study disorder phenomena and the effects of the indistinguishability of the particles in a quantum solid at T = 0 K.14 The Monte Carlo methods for the calculation of multi-dimensional integrals can be used with PIGS and SPIGS because it is easy to show that expectation values computed with the wave function in (1) are formally equivalent to canonical averages of a classical system of special interacting linear open polymers.14,15 This is very similar to what is done in a PIMC simulation, where finite temperature (and not T = 0 K) averages are computed and these are found to be equivalent to canonical averages for a classical system of ring polymers.16 The calculation of expectation values with (1) will differ from the true ground state averages by a quantity that can be reduced below the statistical error of the MC calculation by an appropriate

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choice of δ and the number of projections P , or equivalently of τ = P δ. This is the reason why we can talk about “exact” projection techniques in relation with PIGS and SPIGS, and why with these methods the convergence with τ of the computed expectation values should be always checked. 3. One-Body Density Matrix in Solid 4 He The appropriate Monte Carlo calculations of statistical averages for a Boson system at finite temperature with the PIMC method require the sampling of permutations between particles in order to fulfill the Bose symmetry.16 The fact that without such procedure one cannot recover the correct results can be easily understood by thinking about the classical isomorphism: the sampling of permutations between particles leads to configurations in which more ring polymers join together into a single one; such configurations, which are topologically unreachable without permutations, are of primary importance in computing off-diagonal and superfluid properties because such configurations are those which contribute to give a non-zero superfluid fraction when computed with the winding number estimator.16 The situation is different with the Path Integral projector methods. The projection procedure applied to a wave function which is Bose symmetric, protects the Bose symmetry in Ψτ . This can be easily perceived by thinking about the different classical isomorphism which characterizes these T = 0 K methods: the corresponding classical system is made of linear open polymers; the sampling of permutations between particles corresponds to enable linear polymers to mutually exchange one of their two tails, leaving the system in a configuration which is not topologically different form the starting one and that can be reached by a combination of some (perhaps many) simple moves that involve a single polymer one at a time. While permutations moves are not necessary in SPIGS, it is possible however that such moves in some situation could dramatically improve the efficiency of the sampling procedure by assuring the effective ergodicity of the algorithm. This is why recently we have introduced also their sampling in SPIGS; the sampling scheme that we have implemented in our SPIGS algorithm is the one reported in Ref. 19. We have verified that there are no substantial improvements in the calculations of diagonal properties both in the liquid and in the solid phase; however permutations turn out to be very important for a really efficient estimation of non diagonal properties in the solid phase such as the one body density matrix (OBDM), as we shall discuss below. The OBDM ρ1 (~r − ~r 0 ), at zero temperature, is defined as follows: Z ρ1 (~r − ~r 0 ) = N d~r2 , . . . , d~rN Ψ0 (~r, ~r2 , . . . , ~rN )Ψ0 (~r 0 , ~r2 , . . . , ~rN ) (2) where Ψ0 is the ground state wave function. From the behavior of ρ1 (~r − ~r 0 ), for |~r − ~r 0 | → ∞, which is directly related to the fraction of atoms occupying the zeromomentum one particle state, one can infer whether the system exhibits BEC. The

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o

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(a)

0

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o

|r−r’| [A]

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(b)

Fig. 1. One body density matrix computed along the nears neighbor direction in the basal plane of a 4 He hcp crystal at ρ = 0.0293 ˚ A−3 : (a) commensurate solid (b) solids with fixed concentration of vacancies xv = 1/287.

known property that, at T = 0 K, BEC implies non classical rotational inertia, makes extremely important the evaluation of the OBDM in solid Helium. The reality and positivity of the ground state wave function, but also of its approximation Ψτ , allows one to interpret the integrand in (2) as a canonical weight of a classical system of polymers, one of which has been cut into two halves (we will call them half polymers), one connected to the point ~r and the other connected to the point ~r 0 . Therefore ρ1 turns out to be the histogram of the relative distance of this two points which can be sampled, together with the other coordinates, using a standard Metropolis algorithm. When computing the OBDM in the solid phase, the sampling of configurations in which the two half polymers are far away faces ergodicity problems due to the potential barriers arising from hard core interactions; in fact, the polymers are shown to have a high probability to be rolled up around the equilibrium positions of the lattice. In this framework an accepted permutation, which involves one of the two half polymers with the other polymers, allows it to go beyond, at least partially, an adjacent polymer (4 He atom) and then to overwhelm the potential barriers which slow down the ergodicity of the sampling. During a calculation of the OBDM in solid 4 He at the melting density, this extended SPIGS algorithm is able to sample a permutation involving one half polymer about every 100 MC steps without losing computational efficiency. We have found that a simulation run of about 106 MC step is long enough to observe permutation cycles which involve one of the two half polymers plus a number of other polymers up to 6-7. Permutations between two and three polymers represent the 80% and 16% of the sampled cycles respectively; adding one more polymer to the cycle is found to reduce by an order of magnitude the relative frequency of accepted permutation cycle. In Fig. 3 we show the OBDM computed in a commensurate hcp crystal and in a hcp crystal at fixed vacancy concentration xv = 1/287 at the melting density ρ = 0.0293 ˚ A−3 . The variational estimate of the condensate fraction n0 performed

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with the SWF in the commensurate crystal gave n0 = (5.0 ± 1.7) × 10−6 .20 From evaluations with the “exact” T = 0 K SPIGS method, we have seen that this n0 is remarkably lowered by increasing the number of projection steps toward the true ground state; this presently allow us to provide only an upper bound of about 2.5 × 10−8 for the condensate fraction in the commensurate crystal. We can argue that, in relation with the computation of the OBDM in the commensurate crystal, some correlations are still missing in the variational SWF, resulting in a n0 value far away from the true one, which is recovered only after many projection steps. This limitation of the SWF was not observed when a finite concentration of vacancies was present in the system;21 in this case, in fact, the condensate fraction was shown to converge quickly to the true value after few projections. With the aid of the new extended SPIGS we were able to efficiently study the behavior of ρ1 in large hcp crystals with a finite concentration of vacancies. As shown in Fig. 3(b), the large distance plateau in the OBDM tail clearly indicates the presence of ODLRO in the system with vacancies; and, by keeping xv constant while enlarging the system from 287 to 574 particles, we cannot detect any remarkable size dependence of n 0 . 4. Equilibrium Concentration of Vacancies One of the fundamental questions which has not yet an answer regards the true nature, commensurate or incommensurate, of the ground state of solid 4 He.21 By commensurate we mean that the number of atoms N is equal to the number of lattice sites M , by incommensurate that N 6= M . Present experiments22 do not give any direct evidence for zero point vacancies, i.e. vacancies in the ground state of solid 4 He, but they are able only to put an upper bound of about 0.4%23 on their equilibrium concentration at low temperatures. Neither microscopic computations present in literature do allow to directly answer the question,21 even if arguments against zero point vacancies other than metastable defects have recently appeared. 24 In fact, in Monte Carlo simulations of crystals, periodic boundary conditions (pbc) and lattice structure impose a constraint that makes impossible, in practice, for the system to develop an equilibrium concentration of vacancies.25 This holds not only for canonical and micro-canonical computations, but also for grand canonical simulations.25 Furthermore the equilibrium concentration could be so small to easily have escaped detection in simulations of the ground state of solid 4 He.26 The equilibrium concentration of zero point vacancies xv can then be obtained only indirectly, by a statistical thermodynamical analysis of an extended system, exactly as for classical solids.27 Following Chester’s argument,2 we can infer that translationally invariant wave functions, as the Jastrow wave function (JWF) of the original Chester derivation, describe a solid with a finite xv .21 This argument can be extended also to the SWF and to the SPIGS one. As we have already pointed out, there exists a direct isomorphism between the quantum system at T = 0 K and a suitable classical one at a finite T . It is known that classical solids have a finite equilibrium concentration of

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M = 540 M = 720 −2

10 ρvv(r)

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40

Fig. 2. Intervacancy correlation function ρvv (r) computed with the SPIGS method in two hcp crystals at the melting density containing 3 vacancies. The value of ρvv (r) represent the probability of observing two vacancies separated by a distance r normalized to the multiplicity of the distance r. Each ρvv (r) is obtained considering at least 4 × 106 configurations.

vacancies even if a single vacancy has a finite cost of local free energy because of the gain in configurational entropy.28 Then, given a wave function which describes the ground state of solid 4 He, the problem of computing the equilibrium concentration of vacancies, if any, is formally equivalent to the estimate of xv in the corresponding equivalent classical solid (ecs). The first calculation of xv for solid 4 He based on this quantum-classical isomorphism was done few years ago29 for a JWF with a McMillan30 pseudopotential. However it is known that a JWF is not a good variational wave function for solid 4 He.18 We have now extended this route for computing xv to the SWF, which gives the best variational description of 4 He.18 A preliminary aspect to be investigated is the interaction among vacancies. Recent finite temperature Monte Carlo simulations24 suggest the existence of some kind of attractive interaction, which makes already three vacancies form a tight bound state24 and the vacancy gas instable. With the SPIGS method, we have computed the energy per particle E in systems of increasing size which house an increasing number of vacancies (M = 180, 360 and 540 with 1, 2 and 3 vacancies respectively) in such a way that the concentration of vacancies is preserved in order to rule out non-linear dependence effects on the concentration of vacancies. In the M = 180 case, the pbc constraint the vacancies to be separated by a distance equal to the box side; then, if the attraction among vacancies play a relevant role there should be a gain in E for the large systems where vacancies can be close together: this gain would correspond to the binding energy. Without tail corrections, the obtained energies are E = −4.809 ± 0.003 for M = 180, E = −4.800 ± 0.002 for M = 360 and E = −4.804 ± 0.002 for M = 540 with 1, 2 and 3 vacancies respectively. Due to the chosen geometry for the simulation boxes, the tail corrections turn out to be the same in all the systems. Since the results for E are compatible within the statistical error, we argue that the interaction among vacancies, if any,

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is very weak with a binding energy lower than the statistical error. In order to give a deeper insight into such an interaction, we have studied also the intervacancy correlation function by monitoring their relative distance in the Monte Carlo sampling of incommensurate crystals constructing a radial correlation function ρvv (r). From ρvv (r), plotted in Fig. 4 we see that the vacancies are always able to explore the whole available distance range. Moreover we observe a depletion in the short distance region when enlarging the system size. We may so conclude that, at least in the considered spatial dimensions, no sign of tight bound state is detected. The observed large value of ρvv (r) at nearest neighbor distance is an indication of some short range attraction, but this is not large enough to give a bound state at least for up to 3 vacancies. We conclude that the vacancy gas should be stable in bulk solid 4 He and, for very low concentrations as the expected equilibrium one, the approximation of vacancies as non interacting defects seems to be not so unreasonable. A full grand canonical analysis of an extended classical solid, under the assumption of non interacting defects, leads to an equilibrium concentration of vacancies 27 xv = e−β(µ−f1 ) , where µ is the chemical potential of the perfect crystal, −f1 is the variation in the free energy associated with the formation of one vacancy and β = 1/kB T . Exploiting thermodynamic relations, the exponent in xv can be written as −β(µ − f1 ) = (M − 1)β(f0 − fd ) − βP/ρ. Then, in order to compute xv , we need the free energy per particle βf0 , and the pressure βP/ρ of the perfect ecs, and the free energy per particle βfd , of the defected (i.e. with one vacancy) ecs. Let us consider the ecs of the SWF: it is a solid of triatomic molecules.17,18 The pressure is quite straightforwardly obtained by the virial method31 or from volume perturbations.32 The computation of the free energy (both βf0 and βfd ) is less immediate: it is related to the volume in the configurational space for the ecs (or to the normalization constant for the quantum crystal which is never explicitly computed in a Monte Carlo simulation). In classical systems, the thermodynamic integration method provides a way for reconstructing free energy differences by integration over a reversible path in the phase space.31 For solids this is done by means of the Frenkel-Ladd method33,34 (FLM), whose basic idea is to reversibly transform the solid under consideration into an Einstein crystal. The implementation of FLM is not trivial, we will give more details on our computation method elsewhere.35 We have checked our numerical code for the estimate of the free energy in classical systems with the FLM by reproducing the results in Ref. 34 for a system of soft spheres. A check on the pressure is provided by the convergence to the same value of the two employed methods. With the SWF, the equilibrium concentration of vacancies at the melting density ρ = 0.0293˚ A−3 turns out to be xv = (1.4 ± 0.1) × 10−3 . Since vacancies have been proved to be extremely efficient in inducing BEC in solid 4 He,21 this result would suggest the presence of a finite condensate fraction in bulk solid 4 He, which results then in a supersolid. Moreover, using TBEC for an ideal Bose gas with mass equal to the vacancy effective mass mv = 0.35mHe 36 as an 2/3 estimate of the supersolid transition temperature, we obtain T ' 11.3xv = 141 mK, which is about 2.3 times larger than the experimental transition temperature

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T = 60 mK.12 A possible origin of this small discrepancy is the fact that xv is estimated via a variational technique. Fortunately this route for the estimate of the equilibrium concentration of vacancies xv can be applied also to the “exact” wave function (1) resulting from the SPIGS method. The calculation of xv as a function of τ is under way, we expect that the variational result will be progressively modified under successive imaginary time projections until it reaches its true equilibrium value. 5. Grain Boundaries The study of grain boundaries in solid helium have recently gained a particular interest after the direct experimental observation of superfluidity effects in crystals containing grain boundaries.13 Here we report a preliminary study on such defects in solid 4 He by means of the SPIGS method; this is the first application of an “exact” T = 0 K technique in the study of such systems. We focused on the analysis of crystals containing highly symmetrical grain boundaries: in the system that we will call SGB (from stable grain boundaries), the misorientation between the two crystallites is created by a relative rotation of the two lattices √ around a direction perpendicular to the hcp basal planes (z-axis) of ϑ = 2 arctan( 3/15) ' 13o ; the number of particles (N = 456) √ √ and the dimensions of the simulation box (Lx = √ 2a 57, Ly = a 19, Lz = 2a 6, being a the lattice constant) are chosen in order to match the periodicity of the hcp crystal. The application of pbc in each direction implies the presence of two planar grain boundaries which are perpendicular to the x axis. We have studied also other grain boundaries obtained with different relative rotations around the z-axis (ϑ ' 32o and ϑ ' 38o ) but also around other axis (the y axis with ϑ ' 22o ). The imaginary time parameters used in these simulation was δ = 0.025 K−1 and τ = 0.125 K−1 with the pair-product approximation for the imaginary time propagator; the test on the convergence with τ is under way. Only the SGB system, among those we studied, showed stable grain boundaries as discussed in the following. In our zero temperature simulations, we have observed that the grain boundaries are generally characterized by a high mobility: the positions of the two grain boundaries are found to even move ˚ Angstroms in few thousand of Monte Carlo steps during the simulation. In the SGB system we have observed the two grain boundaries to move always in phase keeping essentially fixed their relative distance that was about 27 ˚ A at ρ = 0.0313 ˚ A−3 . For this reason, in the SGB system, the two crystallites with different orientations have a similar number of 4 He atoms, even if the portion of the simulation cell that they occupy changes during the simulation run. Among the other grain boundaries, those obtained by rotating the lattices around the z axis manifest the same high mobility but, as opposed to the SGB system, the two interfaces show attractive or no correlations. Due to this motion, we found that in few thousand of MC steps, the two grains collapse, leaving in the simulation box a single commensurate crystal. Instead, the system obtained by rotating around the y axis showed a rapid (few tens of thousands of MC

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309 Table 1. Surface energy of a symmetrical tilt grain boundary obtained with a rotation of ϑ ' 13o around the c-axis in an hcp crystal. Density (˚ A−3 )

Surface energy (K ˚ A−2 )

0.0303 0.0313 0.0333 0.0353

0.269 ± 0.008 0.305 ± 0.004 0.406 ± 0.008 0.546 ± 0.005

Surface energy for systems in presence of vacancies (K ˚ A−2 ) 1 vac. 2 vac. 4 vac. 0.244 ± 0.006 0.283 ± 0.004 0.366 ± 0.008 0.482 ± 0.006

GB unstable 0.250 ± 0.005 0.325 ± 0.008 0.418 ± 0.005

GB unstable 0.261 ± 0.005

steps) rearrangement of the atoms toward a highly stressed crystal without grain boundaries. We focused then our simulations on the SGB system. In order to investigate the correlated motion of the two grain boundaries in the SGB system, we have simulated it in a larger box in which the interfaces were placed at different distances in pbc: we found that the two grain boundaries moved in order to maximize their distance in pbc, confirming the repulsive intergrain interaction. The observation at T = 0 K of the high mobility and the correlated motion of the grain boundaries is a surprising effect which is probably connected to a sort of recrystallization waves in systems with such kind of defects. For the SGB system we were able to compute the grain boundary surface energy E with the formula E = (EGB − EP C )N/2A, where EGB is the mean energy per particle of the system in presence of grain boundaries, EP C is the same quantity computed for a perfect crystal in the same simulation box, A is the grain boundary surface and N is the number of particles. The results are shown in Table 1: we notice that the surface energy of the interface increases with the density of the crystal and, at least at the lowest densities, E < 2ELC , being ELC the surface energy of the liquid-crystal interface.37,38 Our results are therefore in agreement with the experimental evidence of the stability of grain boundaries under the condition of phase coexistence between a crystal and a liquid13 and confirm the result obtained from other PIMC calculations at finite temperature.39 We also studied how the SGB system behaves in presence of vacancies. We have found that the activation of a vacancy in the crystal reduces the surface energy of the interfaces (see Table 1). This result can be explained supposing that the vacancies are easily adsorbed into the grain boundary contributing in relaxing the mechanical stress: the energy gain due to this adsorption process increases with the density. These simulation were carried on at fixed lattice constant; the activation of vacancies in the SGB system slightly reduces its average density and we found that, depending on the starting density, the activation of vacancies can destabilize the grain boundaries; such cases are indicated as GB unstable in Table 1. 0 Finally, in these systems, we computed the one body density matrix ρGB 1 (|z−z |) −3 at ρ = 0.0313 ˚ A along a direction parallel to the grain boundary strictly inside 0 the interfacial region. Our results are shown in Fig. 5; the ρGB 1 (|z − z |), computed in the SGB system, starts to oscillate around a non-vanishing value at a distance

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°

|z − z’| [A]

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(a)

(b)

˚−3 along a direction Fig. 3. (a) One body density matrix computed at the density ρ = 0.0313 A orthogonal to the basal planes (z-axis) of a hcp commensurate lattice (dashed line) and along the same direction in the SGB system. (b) snapshot of a configuration of the polymers inside a basal plane of the SGB system; the calculation of ρ1 is along the interfacial regions which are perpendicular to this basal planes.

around 10 ˚ A. The one-body density matrix computed in a commensurate hcp crystal along the same direction orthogonal to the basal planes is a exponentially decaying function of |z − z 0 | in the range we have studied. This seems to suggest that, at zero temperature, also highly symmetrical grain boundaries are defects which can induce in the crystal off-diagonal long range order. At the density ρ = 0.0313 ˚ A−3 , we found that the condensed fraction n0 along these interfaces is of the order of 3 × 10−5 . 0 The tail of ρGB 1 (|z − z |) shows oscillations related to the crystalline structure of the sample: these oscillations match the periodicity of the basal planes and the maxima represent the distance between two corresponding hcp basal planes.

6. Conclusion The evidence is that even if it is possible that supersolid features observed in torsional oscillator experiments find their origin in some non equilibrium phenomena present in the solid 4 He samples, the microscopic studies on this system, by means of Quantum Monte Carlo methods at zero and finite temperature, indicate how subtle is the barrier that separate this system from a macroscopic manifestation of the quantum nature which governs its behavior; it is sufficient to introduce some defects to induce off-diagonal long range order in the system at least locally. Vacancies seems to be the only kind of defects able to induce the coherence of the whole system. The task for the future will be the explanation of the essential mechanism which underlie NCRI in torsional oscillator experiment. Quantum Monte Carlo methods, specific to quantum solids with disorder, will play a leading role to attack this very interesting problem.

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References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39.

A. F. Andreev and I. M. Lifshitz, Soviet Phys. JETP 29, 1107 (1969). G. V. Chester, Phys. Rev. A 2, 256 (1970). A.J. Legget, Phys. Rev. Lett. 25, 1543 (1970); J. Stat. Phys. 93, 927 (1998). M. W. Meisel, Physica B 178, 121 (1992). E. Kim and M.H.W. Chan, Science 305, 1941 (2004); Phys. Rev. Lett. 97, 115302 (2006). N. Prokof’ev, Advances in Physics 56, 381 (2007). E. Kim and M.H.W. Chan, Nature 427, 225 (2004); J. Low Temp. Phys. 138, 859 (2005). A.S.C. Rittner and J.D. Reppy, Phys. Rev. Lett. 97, 165301 (2006). M. Kondo, S. Takada, Y. Shibayama, and K. Shirahama, J. Low Temp. Phys. (in press). A. Penzev, Y. Yasuta, and M. Kubota, J. Low Temp. Phys. (in press). Y. Aoki, J.C. Graves and H. Kojima, Phys. Rev. Lett. 99, 015301 (2007). A.C. Clark, J.T. West, and M.H.W. Chan, arXiv:cond-mat/0706.0906 (2007). S. Sasaki, R. Ishiguro, F. Caupin, H.J.Maris, S. Balibar, Science 313, 1098 (2006). D.E. Galli and L. Reatto, Mol. Phys. 101, 1697 (2003); J. Low Temp. Phys. 134, 121 (2004). A. Sarsa, K.E. Schmidt and W.R. Magro, J. Chem. Phys. 113, 1366 (2000). D.M. Ceperley, Rev. Mod. Phys. 67, 279 (1995). S. A. Vitiello, K.Runge and M.H. Kalos, Phys. Rev. Lett. 60, 1970 (1988); T. MacFarland, S.A. Vitiello, L. Reatto, G.V. Chester and M.H. Kalos, Phys. Rev. B 50, 13577 (1994). S. Moroni, D.E. Galli, S. Fantoni and L. Reatto, Phys. Rev. B 58, 909 (1998). M. Boninsegni, J. Low Temp. Phys. 141, 27 (2005). D.E. Galli, M. Rossi and L. Reatto, Phys. Rev. B 71, 140506(R) (2005). D.E. Galli and L. Reatto, Phys. Rev. Lett. 96, 165301 (2006). R.O. Simmons, J. Phys. Chem. Solids 55, 895 (1994). R.O. Simmons and R. Blasdell, APS March Meeting 2007, Denver, Colorado. M. Boninsegni, A.B. Kuklov, L. Pollet, N.V. Prokof’ev, B.V. Svistunov and M. Troyer, Phys. Rev. Lett. 97, 080401 (2006). W.C. Swope and H.C. Andersen, Phys. Rev. A 46, 4539 (1992). P.W. Anderson, W.F. Brinkman and D.A. Huse, Science 310, 1164 (2005). S. Pronk and D. Frenkel, J. Phys. Chem. B 105, 6722 (2001). C. Kittel, Introduction to solid state physics (Wiley, New York, 1976). J. A. Hodgdon and F. H. Stillinger, J. Stat. Phys. 79, 117 (1995). W. L. McMillan, Phys. Rev. 138, A442 (1965). D. Frenkel and B. Smith, Understanding Molecular Simulations, 2nd Ed. (Academic Press, London, 2002). E. de Miguel and G. Jackson, J. Chem. Phys. 125, 164109 (2006). D. Frenkel and A.J.C Ladd, J. Chem. Phys. 81, 3188 (1984). J.M. Polson, E. Trizac, S. Pronk and D. Frenkel, J. Chem. Phys. 112, 5339 (2000). M. Rossi, E. Vitali, D.E. Galli and L. Reatto, to be published. D.E. Galli and L. Reatto, J. Low Temp. Phys. 124, 197 (2001). S. Balibar, H. Alles and A.Y. Parshin, Rev. Mod. Phys. 77, 317 (2005). F. Pederiva, A. Ferrante, S. Fantoni and L. Reatto, Phys. Rev. Lett. 72, 2589 (1994). L. Pollet, M. Boninsegni, A.B. Kuklov, N.V. Prokof’ev, B.V. Svistunov and M. Troyer Phys. Rev. Lett. 98, 135301 (2007).

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LIQUID 4 HE INSIDE (10,10) CARBON NANOTUBES M. C. GORDILLO Departamento de Sistemas F´ısicos, Qu´ımicos y Naturales. Universidad Pablo de Olavide. Sevilla, Spain ∗ E-mail: [email protected] www.upo.es J. BORONAT and J. CASULLERAS Departament de F´ısica i Enginyeria Nuclear, Universitat Polit` ecnica de Catalunya, Barcelona, Spain. Diffusion Monte Carlo calculations on the adsorption of 4 He inside carbon nanotubes are presented. We focused on the (10,10) ones, in which a ground state liquid phase at low densities and pressures around zero. In those calculations, all the carbon atoms where considered individually, both in the trial function and in their contribution to the total potential. Comparisons are made with the liquid state of helium inside a (5,5) carbon nanotube. Keywords: Carbon nanotubes; liquid 4 He.

1. Introduction One of the most interesting capabilities of carbon nanotubes and their bundles is their purported use as reservoirs, in particular for the lightest quantum fluids. Several adsorption sites were considered as possibilities to accommodate helium atoms and hydrogen molecules, but so far, the most probable locations are the inside of the tubes themselves (when they are open), or the outside part of the assembly of cylinders called nanotube bundles. In this work, we considered what happens to 4 He inside the most common of these nanotubes, the (10,10). We present here Diffusion Monte Carlo calculations on a single tube, since the influence of other tubes in a bundle on a give one could be easily calculated perturbatively. For these tubes, we considered the individual contributions of all the carbon atoms both to the potential and to the trial function. This was so because in a calculation of 4 He inside a (5,5) nanotube some differences were found between a full calculation and one performed with a smoothed out nanotube-helium potential.

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2. Method The trial wave function needed in the diffusion process was chosen to be of the form: Ψ(r1 , . . . , rN ) =

N Y i<j

fHe−He (rij )

N,N Yc

i,j=1

fHe−C (rij )

N Y

Φ(ri )

(1)

i=1

with N and Nc the number of 4 He and C atoms, respectively, and ri the radial distance of particle i to the center of the nanotube. The first two terms of Ψ in Eq. (1) correspond to two-body correlation factors which account for dynamical correlations induced by the He-He and He-C potentials. Both fHe−He (r) and fHe−C (r) are of McMillan type, f (r) = exp[−0.5(b/r)5 ]. In the (10,10) tube case, the lowest energy was found when bC−He = 0, but the narrowness of the environment implies this is not so for the (5,5) tube (bC−He = 2.8 ˚ A). On the other hand, Φ(ri ) = 1 for the (5,5) tube, and has a single maximum at the minimum of the collective C-He potential minimum in the (10,10) cylinder.1 In the case of the narrow tube and the smoothed-averaged C-He potential, we have bC−He = 0 and Φ(ri ) and Gaussian peaked at the center of the tube.2 Further indications about the method and the potentials used are given in Ref. 1. 3. Results Fig. 1 gives us the comparison for the results in the (5,5) tube in the corrugated and smoothed out cases. Full squares indicate the simulation results in the corrugated case, fitted to a third-degree polynomial given by the dashed line. The full line corresponds to the data of Ref. 2, using a non-corrugated tube of the same diameter. The energy scale has been changed to fit in the same figure, since the adsorption energy at infinite dilution changes slightly from the smooth (−429.966 ± 0.001 K) to the corrugated (−429.509 ± 0.001 K) case. We displayed there the difference between the total energy per helium atom and that binding energy. The results of the respective fits indicate that the position of the energy minimum is unaltered (0.0021 ± 0.0001 ˚ A−3 ) in both cases, but the relative binding energy with respects to the limit of infinite dilution changes appreciably (−0.018 and −0.013 K for smooth to corrugated). Since this probably means that the corrugation of the tube makes it to be more closer to a one-dimensional system, in which the relative binding energy is closer to zero.2 Fig. 2 is the low-density part of the ground state isotherm of 4 He inside a (10,10) tube. Such liquid phase is stable up to densities of 0.02 ˚ A−3 ,1 to be replaced at high densities for a phase with a solid helium layer close to the nanotube wall. The liquidness of the phase is established since the trial function in Eq. (1) does not contemplate the atoms to be localized at any particular position. A similar third polynomial fit than for the case of the (5,5) tube indicates that the equilibrium density at zero pressure is 0.0086 ± 0.0001 ˚ A−3 and the corresponding binding energies at infinite dilution and at the minimum are 185.01 ± 0.02 and 187.18 ± 0.02 K, respectively. Obviously, the helium density at the minimum is higher than

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0.04 0.03 Energy per He atom (K)

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0.001

0.002

0.003

0.004

−3

Density (Å ) Fig. 1. Difference between the total adsorption energy and the the binding energy for a single helium atom in a (5,5) tube. Full line, results for a smooth C-He potential, full squares and dashed line, same for a corrugated C-He interaction.

in the (5,5) tube, and the binding energy for a single helium atom is lower. This is due to the fact that the helium atoms are closer to less carbon atoms in the carbon nanotube wall. However, the curvature of the inner part of the cylinder allows the corresponding interactions to produce a bigger energy than in the case of a flat substrate as graphite (binding energy 143 ± 2 K). The greater density is due to the fact that the wider tube allows the helium atoms to form a single liquid layer close to the wall instead of forcing them to stay in a quasi-one dimensional row. However, the curvature of the walls still make the corresponding two-dimensional density at equilibrium (2.9 · 10−2 ˚ A−2 , lower than the corresponding density in a 3 pure two dimensional system (4.4 · 10−2 ˚ A−2 ).

4. Conclusion Diffusion Monte Carlo calculations on 4 He inside two kinds of carbon nanotubes were presented. The results for the liquid ground state indicated that the wider the tube, the lower the helium binding energy, and the higher the equilibrium density at zero pressure. We tested also that the smoothed averaged potential in the narrow tube is adequate but gives slightly different figures in the case of the adsorption energies.

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−182 −183 Energy per He atom (K)

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−184 −185 −186 −187 −188 0

0.004

0.008

0.012

−3

Density (Å ) Fig. 2. text.

Same than in Fig. 1, but for a corrugated (10,10) tube. See further explanation in the

Acknowledgments We acknowledge partial financial support from the Spanish Ministry of Education and Science (MEC) (grants FIS2006-02356-FEDER and FIS2005-04181), Junta de Andalucia (group FQM-205) and Generalitat de Catalunya (grant 2005GR-00779). References 1. M.C. Gordillo, J. Boronat and J. Casulleras, Phys. Rev. B 61, R878 (2000). 2. M.C. Gordillo, J. Boronat and J. Casulleras, Phys. Rev. B 76, 193402 (2007). 3. M.C. Gordillo and D.M. Ceperley, Phys. Rev. B 58, 6447 (1998).

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SPATIAL MICROSTRUCTURE OF FCC QUANTUM CRYSTALS M. J. HARRISON1 and K. A. GERNOTH2 1,2

2

School of Physics and Astronomy, The University of Manchester Oxford Road, Manchester, M13 9PL, UK 1 E-mail: [email protected]

Institut f¨ ur Theoretische Physik, Johannes-Kepler-Universit¨ at, A–4040 Linz, Austria 2 E-mail: [email protected]

Using classical and quantum Fourier Path Integral Monte Carlo (FPIMC) simulations the microscopic one- and two-body densities of fcc rare gas crystals are computed. This is made possible by a thorough formal analysis of the crystallographic point and space group symmetries in these quantities. We present formal and numerical results that are typical of the approach and of the physical systems studied within this framework. Keywords: One- and two-body density; fcc symmetries.

1. Introduction The local one- and two- body densities %(x) and ρ2 (x1 , x2 ) are computed by Monte Carlo methods. Where the system of interest forms a crystalline solid, one may make use of the inherent symmetries of the crystal to improve the efficiency of the simulation or, conversely, one could use the results to confirm the crystalline structure of the system. The work presented here1 is concerned with Face-Centered Cubic (fcc) crystals and builds on earlier work2 for Hexagonal Close-Packed (hcp) crystals. Calculations are undertaken for the Fourier transform ρ(K) of the onebody density and for the various components of the Fourier transform of the two(rot) body density, hγ (r, θ; K), where r is the relative distance and θ the polar angle of x1 − x2 in a rotated x0 y 0 z 0 -frame with its z 0 -axis parallel to the reciprocal lattice vector K. The crux of our approach lies in employing an asymmetric unit (AS) in reciprocal space that by definition, when subjected to the rotations of the point group of the fcc space group, generates the whole reciprocal space. It is useful to first define the basis vectors a1 , a2 , and a3 of the fcc Bravais lattice and the basis vectors of the reciprocal lattice b1 , b2 , and b3 as a a a a1 = (1, 0, 1) , a2 = (0, −1, 1) , a3 = (1, −1, 0) , (1) 2 2 2 b1 =

2π (1, 1, 1) , a

b2 =

2π (−1, −1, 1) , a

b3 =

2π (1, −1, −1) , a

(2)

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where a3 is the volume of the conventional cell and ρ0 = 4/a3 the bulk number density of the crystal. We employ an asymmetric unit AS in reciprocal k-space, for which kx ≥ 0, ky ≤ 0, and kz ≥ 0 and that is bounded by the three planes αK 1 + βK 2 , αK 1 + βK 3 , and αK 2 + βK 3 . The K 6= 0-vectors in AS are given by 2π (j, k, l) , (3) K= a K1 =

2π j (1, −1, 0) , a

K2 =

2π j (1, −1, 1) , a

K3 =

2π j (1, 0, 0) , a

2π 2π 2π (j, −j, l) , K5 = (j, k, 0) , K6 = (j, k, −k) , a a a subject to −j < k < 0, 0 < l < j for K 4 , 0 < l < −k for K, and K4 =

j = 1, 2, 3, · · · ,

j − k, j + l, l − k all even .

(4) (5)

(6)

The vectors K above are general reciprocal lattice vectors, that are invariant only under the identity element of the point group, whereas all other K-vectors listed above are high-symmetry reciprocal lattice vectors which possess invariance groups that are not just the trivial group. 2. One-Body Density The one-body density %(x) is computed via its Fourier transform ρ(K), X ρ(K)eiKx . %(x) =

(7)

K

Applying all rotations Rj of the point group of the fcc space group to AS generates all of reciprocal space. Here we use Rj to denote the 48 rotation matrices of the point group of the fcc space group. The one-body density can be expressed in symmetrized form as 48 23 X ρ(K) X X ρ(K) X ei(Rj K)x = 2 cos ([R2j+1 K]x) , (8) %(x) = g0 (K) j=1 g0 (K) j=0 K∈AS

K∈AS

where we have ordered the rotations in such a way that R 2j K = −R2j−1 K. g0 (K) is the number of rotations that send K into itself. In the Monte Carlo simulations the Fourier transform of the one-body density is sampled once for every attempted move of a particle k using the following relation io n h io n h (new) (old) ρ˜(K) = P exp −iK r k − S (new) + (1 − P ) exp −iK rk − S (old) , cm cm (9) where old and new represent the old and new configurations and S cm is the center of mass of all particles. P is the probability of an attempted move being accepted. The final form of ρ(K) is given by ρ(K) = ρ0 ρ˜(K)/Nconfig , where Nconfig is the number of configurations sampled in the simulation.

(10)

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3. Two-Body Density The two-body density is the joint probability density of finding a particle at x1 and another at x2 . As with the one-body density it is through the Fourier transform that the symmetries are expressed. The two-body density may be written as X u(r; K)eiKS , (11) ρ2 (S, r) = K

where we employ a change of variables such that r = x1 − x2 and S = The Fourier transform can be represented as a series ∞ X urot (r 0 ; K) = h(rot) (r, θ; K)eiγφ , γ

1 2

(x1 + x2 ).

(12)

γ=−∞

where we now write the functions in a rotated x0 y 0 z 0 -frame such that the z 0 -axis is parallel to K. The angles θ and φ are the polar and azimuthal angles of r 0 in the rotated frame and r = |r| = |r 0 | the relative distance. There are two fundamental symmetries associated with the two-body density, namely that ρ2 (S, r) is real and that it is symmetric under exchange of particles, ρ2 (x1 , x2 ) = ρ2 (x2 , x1 ). The symmetries of the fcc space group impose further that the two-body density must be symmetric under inversion, ρ2 (x1 , x2 ) = ρ2 (−x1 , −x2 ). These symmetries, when applied to the functions urot (r 0 ; K), lead to the following relations urot (r 0 ; K) = urot (−r0 ; K) = u?rot (r0 ; K) = urot (r 0 ; −K),

(13)

(r, π − θ; K) = (−1)γ h(rot) (r, θ; K), h(rot) γ γ

(14)

where u?rot (r0 ; K) is the complex conjugate. These relations, when applied to the series (12), yield (rot)

(r, θ; K). h−γ (r, θ; K) = h?(rot) γ 0

(15) 0

Applying the function transformation operators P (R ) to urot (r ; K), where the R0 are the rotations that send K into itself, we find for the fcc space group P (R0 )urot (r0 ; K) = urot R0−1 r0 ; K = urot (r 0 ; K). (16)

This means that urot (r0 ; K) has the same symmetries as K. Symmetrizing for the various high-symmetry classes of K the expression (12) w.r.t. the invariance groups (rot) of K reveals that only certain hγ are non-vanishing with the others forming patterns of characteristic extinctions. These formal results are listed in Table 1. (rot) The component functions hγ (r, θ; K) are sampled according to n o n o 0(new) (new) H(r, θ, γ : K) = P exp −iKS kj exp −iγφkj o o n n (old) 0(old) , (17) exp −iγφkj +(1 − P ) exp −iKS kj (rot)

where H(r, θ, γ : K) is the unnormalized hγ (r, θ; K), S 0kj = S kj − S cm , where S kj is the center of mass of particles k and j, and φkj is the azimuthal angle of vector r 0kj .

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Selection Rules for hγ . (rot)

Non-zero Components hγ

,n∈N

Real {0, 2n} Real {0, 6n}, Imag {6n − 3} Real {0, 4n} Real {0, 2n}, Imag {2n − 1} Real {0, n} Real {0, 2n}, Imag {2n − 1}

K1 K2 K3 K4 K5 K6

4. Results Classical and FPIMC simulations have been performed for solid fcc helium3 at 300 K and 0.1527 ˚ A−3. We have computed ρ(K) for the 10 lowest-lying non-zero K-vectors in the asymmetric unit as well as for all distinct rotations of these vectors. The results yield identical ρ(K) for symmetry-equivalent vectors in agreement with our formal calculations. Numerical results for the two-body density are also in exact agreement with the theoretical symmetry patterns. Shown in Fig. 1 is the 0.2 145° 35° 90°

0.1 ℑ{hγ(r,θ;K)}

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0

-0.1

-0.2 0

Fig. 1.

1

2

3 r [ Å]

4

5

Component Function = {h3 (r, θ; K 2 )}.

imaginary part of the component function h3 (r, θ; K 2 ) for the angles θ under which nearest neighbors appear. As predicted by Eq. (14) we see that this function is antisymmetric w.r.t. θ and π − θ. References 1. M. J. Harrison, Monte Carlo Studies of the Spatial Microstructure of Quantum Rare Gas Crystals, preprint of PhD Thesis submitted to the University of Manchester (2007). 2. K. A. Gernoth, Ann. Phys. (N.Y.) 285, p. 61 (2000); Ann. Phys. (N.Y.) 291, 202 (2001); Z. Kristallogr. 218, p. 651 (2003). 3. H. K. Mao et al., Phys. Rev. Lett. 60, p. 2649 (1988).

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NUCLEATION OF VORTICES IN SUPERCONDUCTORS IN CONFINED GEOMETRIES W.M. WU, M.B. SOBNACK∗ and F.V. KUSMARTSEV Department of Physics, Loughborough University Loughborough LE11 3TU, United Kingdom ∗ E-mail: [email protected] We investigate the nucleation of vortices in a small superconducting disk. We formulate the Gibbs free energy of the disk with an arbitrary number of vortices (arranged in rings concentric with the disk) as a function of temperature and applied magnetic field and minimize the energy to obtain the optimal position of vortices for different applied fields and temperatures. We also analyze the stability of the different vortex states inside the disk and compare our results with those of other theoretical studies and with available experimental observations. Our results are in very good agreement with experiments. Keywords: Vortices; superconductors.

1. Introduction The nucleation of vortices and anti-vortices in superfluid films and ”small” superconducting samples as a function of flow velocity (superfluids) and applied magnetic field (superconductors) has been the subject of many studies in recent years. It is well known that the characteristics of these systems, as well as the nucleation of vortices in the systems,1 are affected by the size and the geometry of the sample. As the size of the system becomes small, the physical boundary conditions become important and the characteristics of vortex interactions in, say, mesoscopic-sized samples are different from those in large samples. Geim and co-workers2–4 found that there could be paramagnetic Meissner effect in a small superconductor. They also investigated the relationship between the applied magnetic field and the free energy near phase transitions. Chibotaru et al.5 and Mel’nikov6 studied the possibility of anti-vortices and multi-quanta vortices penetrating a thin mesoscopic square sample. Schweigert et al.7 concluded theoretically that the multi-vortex state would transform into a single giant vortex state, as the magnetic field and the disk thickness increase. On the other hand, Okayasu and co-workers8 reported that the giant vortex could not be observed in their experiments. While most recent theoretical studies9–14 can indeed explain the creation of different vortex states and their stability very well, they fail to explain some of the rather important and detailed experimental observations. In a series of experiments,

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Grigorieva and co-workers15 studied vortex configurations in mesoscopic superconducting disks as a function of applied magnetic field and found that for a broad range of vorticities L, the vortices form concentric rings or shells — rather like shell filling in atoms and nuclei. They found that the L values (the so-called magic numbers), corresponding to the appearance of new shells, are robust in that they are reproducible in many experiments for different applied fields H and diameter D of their disks. There is up to now no theory which explains the mechanism for vortex shell filling. Theories14 also predict certain stable configurations which are different from those observed experimentally.15 It is appropriate to remark here that while experiments are performed at finite T 6= 0 K temperature, most theoretical studies are strictly only valid at T = 0 K. In this paper, we study the nucleation of vortices in a thin mesoscopic superconducting disk. We extend the preliminary work of Sobnack et al.9 to include temperature by taking into account the entropy associated with the configuration of vortices. Using London’s equation, we write down the free energy of the disk with an arbitrary configuration of vortices and/or anti-vortices arranged in rings. The free energy is minimized and the optimal configurations of vortices are then obtained. We also look at how vortex states evolve inside the disk as a function of applied field and temperature. 2. Formulation and Methodology Consider a small disk with radius R and thickness d in an applied magnetic field H = Hk = ∇ × Aapp , perpendicular to the plane of the disk. We restrict the study to the case R < Λ = λ2 /d (λ is the usual London penetration depth), and d Λ, with H near the lower critical field Hc1 . We further assume that d 6 rc (where rc is the radius of the vortex core), so that the sample can be treated as a 2D plane. We use cylindrical coordinates (r, θ, z) to simplify our calculations and the method we use is essentially that of Sobnack,9 Buzdin10,11 and Baelus.12,14 The restriction on R implies that the screening effects of the superconducting currents js are suppressed: the whole disk can be thought of as soaking in the applied field, so that the local magnetic field B = ∇ × A is approximately the same inside and outside the sample, B ≈ H. We formulate the problem at a temperature T < Tc , where Tc is the critical temperature and we assume that T is small compared with Tc to avoid too much thermal fluctuations inside the disk. At T = 0 K, the Gibbs free energy is made up of the magnetic potential energy together with the kinetic energy of the superconducting currents, Z 1 (B − H)2 + λ2 (∇ × B)2 d3 r. (1) G= 8π

If the applied field H is smaller than the first critical field Hc1 , the sample will attempt to oppose H (the Meissner effect). For H > Hc1 , vortices will penetrate into the disk. For our 2D system, the magnetic field due to the vortices is logarithmic

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and the magnetic flux φv due to each vortex is quantized, Z φv = B(r)r dr dθ = qφ0 ,

(2)

where q ∈ Z (q > 0 for vortices and q < 0 for anti-vortices) and φ0 = hc/2e is the flux quantum. The circulating superconducting current around each vortex is jv = |jv | ∝ 1/r. In the absence of vortices inside the disk, the superconducting current follows from London’s equation and is given by js = −(c/4πλ2 )A. Consider vortices of flux φi = qi φ0 , qi ∈ Z, located at points ri in the plane of the disk. Then London’s equation for the superconducting current is modified to js = −

c (A − Av ), 4πλ2

(3)

where Av ∝ jv is the magnetic potential due to the vortices. 2 For a small disk, the screening is weak and the magnetic potential term (B−H) 2 2 is relatively small compared to the kinetic energy λ (∇ × B) . Then one can write A ' Aapp , and this, together with Eq. (1), leads to Z d (Aapp − Av )2 d2 r (4) G= 8πλ2 for the free energy of the disk. The correct boundary condition on the edge of the disk is that the current js should have only a tangential component. This is achieved for this 2D system, in analogy with electrostatics, by adding for each vortex of flux φi at ri in the disk an image anti-vortex of flux −φi at r0i = (R/ri )2 ri beyond the disk and leads to X Av = [Φi (r − ri ) − Φi (r − r0i )] θˆ (5) i

with Φi (r) =

φi · 2πr

It is easy, though lengthy, to show that, at T = 0 K, Eq. (4) gives the dimensionless energy of a configuration of N1 vortices at ri = r1 , N2 vortices at ri = r2 , each of flux φ = qφ0 , and a vortex (anti-vortex) of flux φ = Lφ0 (L > 0 for a vortex and L < 0 for an anti-vortex) at the centre as g(L, N1 , N2 ) =

1 2 R h + L2 ln − 2LN1 q ln z1 − 2LN2 q ln z2 − Lh 4 rc + g 0 (N1 , 0) + g 0 (N2 , 0) + g12 (N1 , N2 ),

(6)

where g = 16π 2 λ2 G/dφ20 , h = HπR2 /φ0 is the dimensionless applied field and zi = ri /R (i = 1, 2). g 0 (N1 , 0) and g 0 (N2 , 0) are the dimensionless free energies of

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N1 and N2 off-centre vortices respectively, with R − Ni (Ni − 1)q 2 ln zi + Ni q 2 ln(1 − zi2 ) − Ni qh(1 − zi2 ) rc N i −1 X 1 1 − 2zi2 cos(2πn/Ni ) + zi4 2 + Ni q (i = 1, 2), (7) ln 2 4 sin2 (πn/Ni ) n=1

g 0 (Ni , 0) = Ni q 2 ln

where, as is usual, we have introduced the core radius rc as a cutoff whenever ri = rj . The interaction energy g12 (N1 , N2 ) of the N1 vortices in the first ring (radius r1 ) and the N2 vortices in the second ring (radius r2 ) is g12 (N1 , N2 ) = q 2

NX 1 −1 N 2 −1 X n=1 m=1

ln

1 + z12 z22 − 2z1 z2 cos[α + 2π(n/N1 − m/N2 )] , z12 + z22 − 2z1 z2 cos[α + 2π(n/N1 − m/N2 )]

(8)

where α is the misalignment angle between vortices in the two rings. At finite temperatures T 6= 0 K, one has to take into account the entropy associated with N1 vortices at ri = r1 and N2 vortices at ri = r2 . This gives an additional term −kB T (ln W1 + ln W2 ), where Wi = 2πri /2Ni rc (i = 1, 2), giving the dimensionless free energy as g(L, N1 , N2 , t) = g(L, N1 , N2 )−t(2 ln π+2 ln R/rc −ln N1 −ln N2 +ln z1 +ln z2 ), (9) where t = (16π 2 λ2 /dφ20 )kB T is the dimensionless temperature. The magnetization M of the disk follows from ∂(G + M · H)/∂H = 0 and this gives the reduced magnetization m as m(L, N1 , N2 ) = −∂g(L, N1, N2 , t)/∂h. 3. Simulation Results and Analysis For a given temperature t, we minimize g(L, N1 , N2 , t) with respect to z1 and z2 for a range of applied magnetic fields h, with different L, N1 and N2 . Figures 1 and 2 give our results for t = 0 and t = 0.14 respectively. The parameters d, R, λ are chosen so that our disk corresponds to the sample studied experimentally by Grigorieva et al.15 [Niobium, radius R ≈ 1.5 µm, ξ ≈ 15 nm, d ≈ 0.1ξ, critical temperature Tc = 9.1 K] so that in our dimensionless units the critical temperature is tc = 0.7 and t = 0.14 corresponds to the temperature T = 1.8 K at which the experiments were performed. In the following discussions, brackets with only two entries (L, N ) refer to vortex states with a vortex of flux Lφ0 at the center of the disk and N vortices, each of flux φ0 (i.e., q = 1), on a single ring, whereas those with three entries (L, N1 , N2 ) are configurations with 2 rings, with N1 vortices on the first ring, N2 vortices on the second, again each of flux φ0 , and a vortex of flux Lφ0 at the center. As h increases from zero (see Fig. 1), the free energy of the screening currents increases quadratically with h as expected until the first critical field h1 ∼ 4.9 is reached when a single vortex [(L, N ) = (1, 0)] penetrates the disk at the centre (see Fig. 1). This state persists until the second critical field h2 ∼ 7 is reached at which the single center-vortex is replaced by two off-center vortices [(L, N ) =

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t=0 50

40 Free Energy, g

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(0,2,8) (1,8) (1,7) (1,6) (1,5)

30

20

(0,5) (0,4) (0,3)

10

(0,2) (1,0) (0,0)

0 0

5

10 15 Magnetic Field, h

20

25

Fig. 1. The Gibbs free energy g as a function of the applied magnetic field h for different vortex states at t = 0. The stable configurations are successsively (0, 0) → (1, 0) → (0, 2) → (0, 3) → (0, 4) → (0, 5) → (1, 5) → (1, 6) → (1, 7) → (1, 8) → (0, 2, 8). The states (L, N ) = (1, 5), (1, 6), (1, 7), (1, 8) and (0, 2, 8) (corresponding to total flux 6φ0 , 7φ0 , 8φ0 , 9φ0 and 10φ0 respectively) are more stable than (N, L) = (0, 6), (0, 7), (0, 8), (0, 9) and (0, 10) or (1, 9).

(0, 2)]. As h is further increased, more off-center vortices nucleate on the first ring, forming successively a triangle, a square and a pentagon, until the sixth critical field h6 ∼ 13 is reached. Then it is energetically more favorable for the next vortex to nucleate at the center of the disk [(L, N ) = (1, 5)] than to form a hexagon of six off-center vortices with a vortex at each vertex. This result agrees with those of other studies2,7,11,12 and is also analogous to the result of the study by Yarmchuck and Gordon13 on the nucleation of vortices in superfluid helium, where it was shown that a central vortex would 3 appear in the state: the vortex at the center is very important to stabilize the off-center vortices on the ring. When the number of offcenter vortices is small (N 6 5), the interaction forces between vortices are not large enough to cause instabilities. As h increases further, further off-center vortices enter the disk and nucleate on the first ring, so that for the range of h shown, the stable vortex states and transitions are (L, N ) = (0, 0) → (1, 0) → (0, 2) → (0, 3) → (0, 4) → (0, 5) → (1, 5) → (1, 6) → (1, 7) → (1, 8), until the tenth critical field

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hc10 ∼ 17.5 is reached when the vortex state changes rather dramatically in that instead of the tenth vortex nucleating to form state (1, 9), the vortices rearrange themselves to form the state (L, N1 , N2 ) = (0, 2, 8), with no vortex at the center of the disk, two vortices on the first ring and eight on the second.

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(0,2,8)

30

(0,2,7)

25

(1,7) (1,6) (1,5) (0,5) (0,4) (0,3)

20 15 10 5 0 0

(0,2) (1,0) (0,0)

5


20

25

Fig. 2. The Gibbs free energy g as a function of h at t = 0.14 (T = 1.8 K). The stable vortex states and transitions are (0, 0) → (1, 0) → (0, 2) → (0, 3) → (0, 4) → (0, 5) → (1, 5) → (1, 6) → (1, 7) → (0, 2, 7) → (0, 2, 8) as h increases (for the range of h shown).

At t = 0.14, the Meissner state persists until the applied magnetic field reaches the first critical field h1 ∼ 4.7 when a single vortex [(L, N ) = (1, 0)] nucleates at the center of the disk (see Fig. 2). At the second critical field h2 ∼ 5.2, the energetically favorable configuration (with total flux 2φ0 ) is the state with two off-center single vortices (L, N ) = (0, 2). As h increases further, more vortices penetrate the disk, with the fluxiod state going successively through the transitions (0, 2) → (0, 3) → (0, 4) → (0, 5) → (1, 5) → (1, 6) → (1, 7) until the next critical field h8 ≈ 17.5 is reached. Then an extra vortex enters the disk, but the stable vortex state with total flux 9φ0 is the state (L, N1 , N2 ) = (0, 2, 7): no vortices at the centre of the disk, and the nine vortices form two rings, with two vortices on the inner ring and seven on the outer. As h is further increased, a further vortex penetrates the disk and

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nucleates on the outer ring, forming the state (0, 2, 8). These results are in direct agreement with the experimental observations of Grigorieva et al.15 Figure 3 shows blown up parts of Figs. 1 and 2 giving the Gibbs free energy g as a function of h for values of h for which the total vortex flux in the disk is 9φ0 and Figure 4 the corresponding one for total vortex flux 10φ0 .

t=0

t = 0.14

34

34

33

33

32

32 Free Energy, g

Free Energy, g

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31 30 29 28

31 30 29 28

27

27

26

26

25 10


25 10


Fig. 3. Total flux = 9φ0 . The solid lines and those represented by ∗ represent the Gibbs free energy g of the states (0, 2, 7) and (1, 8) respectively as a function of the applied magnetic field h. The figure on the left is for temperature t = 0 (T = 0 K), while the one on the right is for t = 0.14 (T = 1.8 K).

The solid curves in Fig. 3 gives the magnetic field dependence of the free energy of the vortex state (0, 2, 7) and the curve represented by stars (∗) that of state (1, 8), showing clearly that for configurations with total flux 9φ0 , the most stable state at t = 0 (T = 0 K) is the state (1, 8), with a central vortex and eight vortices on a shell, forming an octet, whereas at t = 0.14 (T = 1.8 K) the most stable state is the state (0, 2, 7), with no vortex at the center of the disk, two vortices on the inner ring and seven on the outer ring. This result is in very good agreement with the experiments of Grigorieva et al.15 (who found that at T = 1.8 K, the state (1, 8) was observed in only just a few cases, while the state (0, 2, 7) was, by far, the most

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frequently observed state) and in contradiction with the studies of Baleus et al. 14 who theoretically predicted only the state (1, 8).

t=0

t = 0.14

39

39

38

38

37

37 Free Energy, g

Free Energy, g

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36 35 34 33

36 35 34 33

32

32

31

31

30 18


30 18


Fig. 4. Total flux = 10φ0 . The dashed lines (−−), the solid lines and the lines represented by stars (∗) are the field dependence of the vortex states representing the states (0, 3, 7), (1, 9) and (0, 2, 8) respectively. The figure on the left is for t = 0 (T = 0 K), while the one on the right is for t = 0.14 (T = 1.8 K).

Figure 4 shows the corresponding result for vortex states with total flux 10φ 0 , with the result for t = 0 on the left and that for t = 0.14 on the right. The dashed curves show the dependence on h of the free energy of the state (0, 3, 7), the solid lines that of the state (1, 9) and the lines represented by stars (∗) that of state (0, 2, 8). At t = 0, the most stable vortex state is the state (0, 2, 8), with states (0, 3, 7) and (1, 9) having almost the same (slightly higher) energy. At t = 0.14 (T = 1.8 K), the state (0, 2, 8) is again the most stable state, with the other two states having higher energies. This result is in very good agreement with the experimental studies of Grigorieva et al.15 who reported that the state (0, 2, 8) was the most frequently observed state and that the state (1, 9) was never observed in their experiments. Baleus et al.,14 on the other hand, predicted only the state (1, 9) for the state with total flux 10φ0 .

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4. Concluding Remarks We have presented an extension of the early study of Sobnack and Kusmartsev 9 by including temperature into the formulation and by considering two-ring vortex states. Inclusion of the temperature term −T S (by taking into account the entropy S associated with the non-center vortices) lowers the free energy of some of the vortex states and stabilizes them. Our results are in very good agreement with those of the recent experiments of Grigorieva and co-workers15 and in contrast with some of the results of Baleus et al.14 Some of the states they theoretically predicted are either only rarely observed or not observed in the experiments.15 The main reason for the disagreement between the experiments of Grigorieva et al.15 and the theory of Baleus et al.14 is that the experiments are performed at finite temperatures T 6= 0 K , whereas the study of Baleus et al.14 is only valid at T = 0 K. We are currently working on extending this work to include the possibility of vortices nucleating in multi-rings (> 2). We are also working on establishing the background physics mechanism responsible for populating each vortex shell (ring), beyond the understanding that the appearance of each new ring is dictated by achieving the lowest energy. References 1. Alan T. Dorsey, Nature (London) 408, 784 (2000). 2. A. K. Geim et al., Nature (London) 390, 259 (1997). 3. A. K. Geim, S. V. Dubonos, J. G. S. Lok, M. Henini, and J. C. Mann, Nature (London) 396, 144 (1998). 4. A. K. Geim et al., Nature (London) 407, 55 (2000). 5. L. F. Chibotaru, A. Ceulemans, V. Bruyndoncx, and V. Moshchalkov, Nature (London) 408, 833 (2000). 6. A. S. Mel’nikov et al., Phys. Rev. B 65, 140503-1 (2002). 7. V. A. Schweigert et al., Phys. Rev. Lett. 81, 2783 (1998). 8. S. Okayasu, T. Nishio et al., IEEE 15(2), 696 (2005). 9. M. B. Sobnack and F. V. Kusmartsev et al., Recent Progress in Many-Body Theories (World Scientific, Singapore, 365 (2006)). 10. A. I. Buzdin, Phys. Rev. B 47, 11416 (1993). 11. A. I. Buzdin and J.P. Brison, Phys. Rev. A 196, 267 (1994). 12. B. J. Baelus and F. M. Peeters, Phys. Rev. B 65, 104515-1 (2002). 13. E. J. Yarmchuk and M. J. V. Gordon, Phys. Rev. Lett 43, 214 (1979). 14. B. J. Baelus, L. R. E. Cabral and F. M. Peeters, Phys. Rev. B 69, 064506-1 (2004). 15. I.V. Grigorieva et al., Phys. Rev. Lett 96, 077005 (2006).

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THE CORRELATED DENSITY AND THE BERNOULLI POTENTIAL IN SUPERCONDUCTORS 4 ´ 3,4 , Jan KOLA ´ CEK ˇ Klaus MORAWETZ1,2 , Pavel LIPAVSKY 1 Institute

of Physics, Chemnitz University of Technology, 09107 Chemnitz, Germany for the Physics of Complex Systems, 01187 Dresden, Germany 3 Faculty of Mathematics and Physics, Charles University, Ke Karlovu 3, 12116 Prague 2 Institute of Physics, Academy of Sciences, Cukrovarnick´ a 10, 16253 Prague 6, Czech Republic E-mail: [email protected] 2 Max-Planck-Institute

4

The kinetic theory with non-instantaneous collisions predicts that the density of quasiparticles differs from the particle density by the correlated density. Similarly the correlated density appears in superconductors where it is visible via the electrostatic potential known as the Bernoulli potential. We discuss the experimental access to this phenomenon. Keywords: Correlated density; nonlocal kinetic theory; Bernoulli potential; superconductivity; NMR; YBCO.

The nonlocal quasiparticle kinetic equation for the momentum-, space-, and timedependent distribution function derived within the method of non-equilibrium Green functions1–3 has the form of a Boltzmann equation, where the collision event is nonlocal and non-instantaneous. Arguments of the distribution functions in the collision integral contain corresponding nonlocal shifts4 f2 ≡ fb (p, r − ∆2 , t), f3 ≡ fa (k − q − ∆K , r − ∆3 , t − ∆t ), and f4 ≡ fb (p + q − ∆K , r − ∆4 , t − ∆t ). The scattering rate is the modulus of the scattering T-matrix P = |T R |2 . All corrections ∆, describing the nonlocal and non-instantaneous collision are given by derivatives of the scattering phase shift φ = Im lnT R (Ω, k, p, q, t, r) p+q ∂φ ∂φ ∂φ ∂φ p , ∆2 = − − ∆t = , ∂Ω ∂p ∂q ∂k ∂φ 1 ∂φ , ∆3 = − , 2 ∂t ∂k ∆ k-q k 1 ∂φ ∂φ ∂φ , ∆4 = − + ∆K = . 2 ∂r ∂k ∂q out (1) Thermodynamical properties implied by the nonlocal kinetic equation cover quantum virial corrections on the binary level. It should be noticed that the nonlocal corrections do not increase the computational time as compared to solving the ‘local’ Boltzmann equation.5,6 In contrast, a numerical implementation of backflows ∆4

∆

∆E = −

2

3

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of the Landau type, which also yield binary virial corrections, brings appreciable complications. The nonlocal kinetic equation conserves density, momentum and energy including the corresponding two-particle correlated parts.2 In analogy with the Boltzmann equation one can define the one-particle quantities like the mean quasiparticle density nqp , the mean momentum Qqp , the mass current j qp and the stress tensor Jijqp in the form P k2 P P ∂ε E qp = ( 2m + 12 σmf )fk , nqp = f, j qp = ∂k f, k k k P P ∂ε Qqp = k f, Jijqp = kj ∂ki + δij ε f − δij E qp , (2) k

k

where ε is the quasiparticle energy and σmf the mean-field part of the selfenergy. The one-particle quantities do not conserve. From the nonlocal kinetic equation the modified balance equations follow as2 ∂(nqp + ncorr ) ∂(j qp + j corr ) + =0 ∂t ∂r qp qp corr corr ∂(Qj + Qj ) X ∂(Jij + Jij ) + =0 ∂t ∂ri i

∂(E qp + E corr ) ∂E = = 0. ∂t ∂t

(3)

Briefly, the conserving observables consist of the sum of the quasiparticle parts (2) and the correlated parts which we can interpret as a contribution of correlated pairs Z Z Z Z k+p k + p ncorr = dP∆t , j corr = dP∆3 , Qcorr = dP ∆t , E corr = dP ∆t , 2 2 Z 1 dP kj ∆3i + pj (∆4i − ∆2i ) + qj (∆4i − ∆3i ) . (4) Jijcorr = 2 Here dP = δ1234 |tsc |2 f1 f2 (1 − f3 − f4 )dp1 dp2 dp3 dp4 /(2π)8 reminds the probability to form a pair per unit of time. Note that the factor 1 − f3 − f4 can be negative what signals anti-correlations between particles. The density of pairs is given, if dP is multiplied by the lifetime of the pair ∆t . The current and momentum curried by pairs result if we multiply the pair density by the velocity ∆3 /∆t and the momentum 1 2 (k+p), respectively. The correlated part of the stress tensor is known in the physics of gases as the collision flux. The nonlocal corrections with parameters found from collisions of two isolated nucleons7 have been implemented into numerical simulations of heavy ion reactions in the non-relativistic regime.5,6 The resulting temperature of mono-nucleon products of central reactions increases due to nonlocal effects towards experimentally observed values, while the temperature of more complex clusters remains unchanged.5 The experimental proton distribution in very peripheral reactions could be described in this way.6

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It is a pity that the concept of the correlated density did not find its implementation within the solid state theory yet. Let us demonstrate its use for superconductors. In superconductors the correlated density equals the difference of electron densities in the superconducting and the normal states for a fixed Fermi momentum. Within the BCS theory at the zero temperature one finds8 2ωcut ∂N ∆2 ncorr ≈ ln √ , (5) ∂EF 4π e∆

where ∆ is the BCS gap and ωcut is the cut-off frequency. The correlated density appears only if the particle-hole symmetry is disturbed being proportional to the slope of the density of states at the Fermi energy ∂N/∂EF . Since the total system stays (quasi)neutral, for an inhomogeneous gap the correlated density is compensated by changes in the normal part of density. This is achieved by space variation of the Fermi momentum, δnqp = N δEF . As the electrochemical potential µ = eϕ+EF is constant in equilibrium, the electrostatic potential compensates changes of the Fermi energy, i.e., it reads 2ωD 1 ∂N ∆2 √ . (6) ln eϕ = N ∂EF 2 e∆

While the correlated density is rather a theoretical concept, the electrostatic potential is an observable physical quantity. To be able to test correlations via the electrostatic potential a number of additional realistic features have to be included, however. An inhomogeneity of the gap in a superconductor appears due to diamagnetic currents, therefore it is necessary to take into account super-currents and magnetic fields. Moreover, it is desirable to assume finite temperatures. Both features are covered by phenomenological theories developed during the late sixties. 9 Near the critical temperature Tc it is convenient to describe the gap with the 1 2 2 Ginzburg-Landau theory 2m ∗ (−i~∇ − 2eA) ∆ + α (T − Tc )∆ + β |∆| ∆ = 0, where parameters α and β depend on the Fermi energy via the density of states. The vector potential results from the Ampere law ∇2 A = j with the super-current j = (e/m∗ )n|∆/∆0 |2 (2eA + ~∇χ), where ∆ = |∆|eiχ and ∆0 is the gap at zero temperature. Once the gap is solved, the electrostatic potential is easily derived from the Ginzburg-Landau free energy as the condition of stability with respect to the charge transfer.10 The resulting electrostatic potential has a part proportional to the kinetic energy of the super-current for which it is also called the Bernoulli potential. Measurements of the Bernoulli potential11,12 via the Kelvin method have shown that the contribution of the particle-hole non-symmetry ∝ ∂N/∂EF is absent in the work function. This has been explained only recently by a surface dipole.13 The surface dipole obtained from the Budd-Vannimenus theorem14 exactly cancels these contributions under conditions in the above mentioned experiments. To avoid the compensation via the surface dipole, it is desirable to make measurements directly in the bulk. While the electrostatic potential is not accessible

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by recent methods, it is possible to observe the related charge transfer. Kumagai et al 15 have measured the nuclear magnetic resonance lines in the high-Tc material YBaCuO. From the dependence of the quadrupole resonance on the applied magnetic field it was possible to identify the charge transfer between CuO2 layers and CuO chains caused by a suppression in the cores of vortices. An agreement with the particle-hole non-symmetry mechanism was found.16 Summarizing, the nonlocal kinetic theory unifies the Landau theory of quasiparticle transport with the theory of dense quantum gases. The balance equations for the density, momentum and energy include quasiparticle contributions and the correlated two-particle contributions beyond the Landau theory.17 Implementations of nonlocal collisions to heavy-ion reactions improves the agreement between the predicted and the observed productions of particles. The heavy-ion reactions provide only indirect tests of theoretical concepts employed. To support more direct experimental studies of this concept we have discussed the correlated density in superconductors and have shown that it can be observed via the Bernoulli potential and the related charge transfer. Acknowledgment This work has been supported by DAAD in Germany and by the research program MSM0021620834 of the Ministry of Education of Czech Republic. References ˇ cka, P. Lipavsk´ 1. V. Spiˇ y and K. Morawetz, Phys. Lett. A 240, p. 160 (1998). ˇ cka, Kinetic equation for strongly interacting 2. P. Lipavsk´ y, K. Morawetz and V. Spiˇ dense Fermi systems, Annales de Physique, Vol. 26,1 (EDP Sciences, Paris, 2001). ˇ cka, Ann. of Phys. 294, p. 134 (2001). 3. K. Morawetz, P. Lipavsk´ y and V. Spiˇ ˇ cka, K. Morawetz and P. Lipavsk´ 4. V. Spiˇ y, Phys. Rev. E 64, p. 046107 (2001). ˇ cka, P. Lipavsk´ 5. K. Morawetz, V. Spiˇ y, G. Kortemeyer, C. Kuhrts and R. Nebauer, Phys. Rev. Lett. 82, p. 3767 (1999). 6. K. Morawetz, P. Lipavsk´ y, J. Normand, D. Cussol, J. Colin and B. Tamain, Phys. Rev. C 63, p. 034619 (2001). ˇ cka and N.-H. Kwong, Phys. Rev. C 59, p. 3052 7. K. Morawetz, P. Lipavsk´ y, V. Spiˇ (1999). 8. K. Morawetz, P. Lipavsk´ y, J.Kol´ aˇcek, E. H. Brandt and M. Schreiber, Int. J. Mod. Phys. B 21, p. 2348 (2007). 9. G. Rickayzen, J. Phys. C 2, p. 1334 (1969). 10. P. Lipavsk´ y, J. Kol´ aˇcek, K. Morawetz and E. H. Brandt, Phys. Rev. B 65, p. 144511 (2002). 11. J. Bok and J. Klein, Phys. Rev. Lett. 20, p. 660 (1968). 12. T. D. Morris and J. B. Brown, Physica 55, p. 760 (1971). 13. P. Lipavsk´ y, J. Kol´ aˇcek, J. J. Mareˇs and K. Morawetz, Phys. Rev. B 65, p. 012507 (2001). 14. H. F. Budd and J. Vannimenus, Phys. Rev. Lett 31, p. 1218 (1973). 15. K. I. Kumagai, K. Nozaki and Y. Matsuda, Phys. Rev. B 63, p. 144502 (2001). 16. P. Lipavsk´ y, J. Kol´ aˇcek, K. Morawetz and E. H. Brandt, Phys. Rev. B 66, p. 134525 (2002). ˇ cka and K. Morawetz, Phys. Rev. E 59, p. R1291 (1999). 17. P. Lipavsk´ y, V. Spiˇ

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ELECTRON CORRELATIONS IN SOLIDS: FROM HIGH-TEMPERATURE SUPERCONDUCTIVITY TO HALF-METALLIC FERROMAGNETISM E. ARRIGONI, L. CHIONCEL and H. ALLMAIER Institute of Theoretical Physics and Computational Physics, Graz University of Technology, Petersgasse 16, 8010 Graz, Austria M. AICHHORN and W. HANKE Institute for Theoretical Physics and Astrophysics, University of W¨ urzburg, Am Hubland, 97074 W¨ urzburg, Germany Electron correlations are responsible for a number of remarkable phenomena including exotic magnetic phases, non-Fermi liquid behavior, and high-temperatures superconductivity. These properties cannot be described by mean-field-like methods based on density-functional theory. The aim of this talk is to discuss some of these phenomena, and present results obtained from a recently developed quantum-cluster method, the Variational-Cluster-Approach. Specifically, we will discuss high-temperature superconductors and half-metallic ferromagnets, with emphasis on correlation effects. We will also show how the combination of density-functional and quantum-cluster methods can be used to obtain a satisfactory ab-initio description of these strongly-correlated materials. These notes are a summary of the talk held at the International Conference on “Recent Progress in Many-Body Theories RPMBT14” in Barcelona, containing a short, and by no means complete discussion on the topic of strong correlations as well as some recent works of the author on the subject. For a more complete and appropriate overview, we refer the reader to the extended literature on the subject. Keywords: Strong correlation; high-temperature superconductivity; half-metallic ferromagnets.

1. Introduction: Correlations in High-Temperature Superconductors The discovery of the high-temperature superconductors (HTSC) in 1986 by J. G. Bednorz and K. A. M¨ uller1 certainly triggered the interest and an intense research effort on strongly correlated materials. HTSC are not only interesting because of superconductivity, but especially because they are characterized by a variety of phases which can be obtained by changing doping and temperature. These materials have a quite complicated crystal structure (see Fig. 1), although there is substantial agreement that the important electron dynamics takes place on two-dimensional copper-Oxide (CuO2 ) layers, which are a common characteristics of all HTSC.

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>=1

e

Sr 2+ La

3+

Doping

_

Temperature (K)

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Antiferromagnetic

Insulator

60 40

Super− conductor

20 0.

0.1

0.2

0.3

x = Sr doping Cu2+ O 2−

Fig. 1.

(left) and it schematic phase diagram as a function of doping and temperature (right).

In the undoped case, there is one valence electron per unit cell in the CuO2 planes, which means that according to a band picture the system should be a metal with an half-filled band. This is in contrast to experimental results that clearly show that the material is an insulator with an optical gap of about 2eV. This insulating behavior can be understood by using a local picture: adding an electron (with opposite spin) to a copper d orbital in which there is already another electron costs a large amount of energy U (typically U ≈ 4eV), the so called Hubbard repulsion. The reason is that d-orbitals are rather localized, so the two electrons on d orbitals come very close together. If the energy U is larger than the energy gain obtained by delocalizing the electron wave function, which is proportional to the “hopping” parameter t, then electrons prefer to have localized wavefunctions rather than paying the energy U associated with this “double occupation” process, and the material is an insulator. This kind of insulating behavior, which is different from the conventional band insulator mechanism, was first suggested by N. Mott2 and therefore takes his name (see Ref. 3 for a review). The Mott insulating behavior is perhaps one of the most important striking effects in which the failure of a singleparticle description and, consequently, the importance of correlations is evident. Upon doping, which in the case of La2−x Srx CuO shown in Fig. 1 is obtained by replacing La with Sr, electrons are removed from the CuO2 layers, or, equivalently holes are introduced. The material becomes superconducting above a certain doping x = xmin ≈ 6%, the optimum superconducting transition temperature Tc is achieved at x = xopt ≈ 15%. Eventually, above a certain doping x ≈ 25% the material remains a normal metal down to zero temperatures. For temperatures above the superconducting dome and for x up to xopt , HTSC materials are metals, however with unconventional properties. First of all, several physical properties show a socalled “pseudogap”, secondly the metal cannot be described in terms of conventional Fermi-liquid theory.4

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Below a certain doping (x ≈ 2% in LaSrCuO), HTSC are antiferromagnets . This means that neighboring electrons, which are localized because the material is an insulator, prefer to have antiparallel spins. 2. Model and Methods The interplay between kinetic energy and repulsive energy discussed above is well described by the two-dimensional Hubbard Hamiltonian: X X nR↑ nR↓ , (1) tRR0 c†Rσ cR0 σ + U H= RR0 σ

R

where cRσ , c†Rσ are annihilation and creation operators for electrons with spin σ =↑ , ↓ on a Copper d site R, tRR0 denotes the amplitude for an electron to hop from site R to site R0 , nRσ = c†Rσ cRσ is the number of electrons with spin σ at the copper site R, and U is the local Coulomb repulsion. It is clear that his Hamiltonian is greatly simplified, since other orbital degrees of freedom (e.g., oxygen p orbitals), long-range Coulomb interaction, electron-phonon coupling, etc, are left out. Nevertheless, this choice appears to be legitimate, last not least because of the amazing agreement achieved between numerical simulations and experimental results for the normalstate properties of the cuprates (see, for example, Refs. 3 and 5). 2.1. Variational cluster approach The model (1), although apparently so simple, cannot be solved exactly except in 1 dimension. Therefore, it is important to find an appropriate approximation that correctly takes into account correlation effects. Since, as discussed above, local correlations are important (see, e.g., Ref. 6 for a discussion of this issue), one can start with a small system, i.e., a finite cluster. In a cluster with N sites the dimension of the Hilbert space (neglecting symmetries) is equal to 4N , since each site can have four different configurations. Symmetries such as particle number and spin conservation can be used to decouple different sectors of the Hilbert space and, hence, reduce the effective dimension, which, nevertheless, still grows exponentially with N . However, if the cluster is not too large, one can diagonalize the Hamiltonian matrix corresponding to Eq. (1) numerically. For Hilbert spaces of up to about 10 8 states one can use the Lanczos method to find the low-lying states. The question is how to recover the original infinite lattice. One solution is provided by the so-called cluster-perturbation-theory (CPT) method. Here, one splits the lattice into finite clusters and then treats the intercluster hopping perturbatively.7,8 The Hamiltonian can then be split into a part describing disconnected clusters (Hcl ) and an intercluster (Hintercl ) part containing the hopping processes between the clusters: H = Hcl + Hintercl .

(2)

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To first order in the intercluster Hamiltonian Hintercl , the CPT Green’s function reads (in matrix notation) −1 G−1 CP T = Gcl − T ,

(3)

where Gcl is the exact cluster Green’s function, and T is a matrix describing the intercluster hoppings. The advantage of this approach is that it is (in principle) exact in three limiting cases, that is in the atomic limit (T = 0), or in the noninteracting limit (U = 0), or for the size of the cluster N → ∞. The disadvantage is that spontaneous symmetry-broken phases such as antiferromagnetism (AF) or superconductivity (SC) cannot be addressed whenever the cluster ground state is non degenerate. The reason is that no matter how large the cluster is, the ground state will always have the full symmetry of the problem, and the perturbative treatment Eq. (3) does not change this result. The idea of the so-called Variational Cluster Approach (VCA)9,10 is basically to proceed in a similar way as in standard weak-coupling perturbation theory, namely to break the symmetry at some unperturbed (mean-field) level and then carry out a perturbation expansion around this symmetry-broken unperturbed state. The goal is essentially to start from a “better” unperturbed state that already contains some properties of the fully perturbed state. In our case, this is achieved by adding one (or more) symmetry-breaking fields to the unperturbed Hamiltonian Hcl , given by Hamiltonians of the form X (4) hi Fî , Hf ield = i

where Fî are single-particle operators (for example, the staggered magnetization, and/or the pairing operator with a given symmetry, etc.), and hi the corresponding fields (i.e., coefficients). In this way, one replaces Hcl → Hcl + Hf ield .

(5)

However, since we want to describe spontaneous and not explicit symmetry breaking, the total Hamiltonian H must remain unchanged. This is achieved by subtracting the field Hamiltonian from the “intercluster” part: Hintercl → Hintercl − Hf ield ,

(6)

which now describes the intercluster hoppings plus intracluster fields. Provided Hf ield is a single-particle operator, the expression (3) remains formally the same, although, of course, Gcl and T acquire a dependence on the hi . Clearly, if (3) was exact, the result would not depend on the hi . This is, for example, the case for U = 0. In practice, results do depend on the fields hi and the question is what are the “optimal” values for the hi for a given problem. The first guess would be to find the hi that minimize the appropriate free energy of the system, which in this case corresponds to the grand-canonical potential Ω, as we are working at fixed chemical potential µ. Within the CPT approximation, the grand-canonical potential Ω is

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given by ΩCP T = Ωcl + Tr ln GCP T − Tr ln Gcl ,

(7)

where Ωcl is the grand-canonical potential of the disconnected clusters, which can be determined by exact diagonalization (along with Gcl ). Of course, this is also valid when the “cluster” also contains symmetry-breaking terms as discussed above. This is the VCA, however presented in a simplified, intuitive formulation. A more rigorous framework for VCA is provided by the Selfenergy Functional Approach (SFA) developed by M. Potthoff.11 For a detailed description of the method we refer to Ref. 11. 3. Applications 3.1. Phase diagram of high-temperature superconductors Most HTSC acquire charge carriers by doping with holes, but there are also electrondoped HTSC materials. These electron-doped materials display a phase diagram which is similar to the hole-doped ones, but there are some important differences. One characteristic feature is the fact that in the electron-doped case the AF phase extends to much higher dopings, while the SC phase has a smaller doping extension. In addition, the pseudogap energy and Tc are generically smaller in the electron doped case. In order to study the phase diagram of the HTSC materials in the electronand in the hole-doped case, we have considered the two-dimensional single-band Hubbard model (1) including nearest-neighbor (t) and next-nearest-neighbor (t0 ) hoppings. We have taken typical parameters valid for both hole- and electron-doped high-Tc cuprates,12 that is t0 /t = −0.3 and U/t = 8. We summarize here results presented in previous publications,13–15 see also Ref. 16. For more details, we refer to these works. Since we expect to describe both an AF and a d-wave SC phase, Hf ield contains both a staggered and a nearest-neighbor d-wave pairing field. More specifically, Hf ield = HAF + HSC + H , (8) P 0 where HAF = h R (nR,↑ − nR,↓ )eiQ·R , and HSC ∆ R,R0 η(R − R )(cR,↑ cR0 ,↓ + 2 h.c.), where the conventions are the same as for (1). In addition, Q = (π, π) is the AF wave vector, and the d-wave factor η(R − R0 ) is non vanishing for nearestneighbor lattice sites only and is equal to +1 (−1) for R − R0 in x (y) direction. In the SC term, the sum is restricted to R and R0 belonging to the same cluster. Moreover, away from half-filling, it is necessary to add a “fictitious” on-site energy to the cluster, i.e., X H = nR,σ , (9) P

R,σ

which plays the role of a “shift” in the cluster chemical potential with respect to the “physical” chemical potential µ. Again, this term is “fictitious” because it is

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subtracted via (6). Without this term, the mean particle density n = 1 ∓ x (x is the doping, ∓ corresponds to hole- and electron-doping, respectively) cannot be unambiguously determined, as different results would be obtained by evaluating it as n = −∂Ω/∂µ or as the usual trace over the Green’s function. Physically, one can regard as a Lagrange multiplier which enforces an appropriate constraint in the particle number. Quite generally, different stationary solutions can be found, corresponding, for example, to different phases. In this case, the minimum Ω selects the most stable phase. Near half-filling, the two most stable solutions are a coherently mixed AF+SC and a pure SC phase. Results for the order parameters for these solutions are plotted in Fig. 2 for the hole- and electron-doped cases, for different cluster sizes and shapes as indicated in the figure. The good result is that the order parameters indeed do not show a strong dependence on the cluster size, except close to the critical doping at which they vanish. This means that these results are already quite accurate and that the order parameters should remain finite in the exact solution (which is ideally achieved when the clusters become infinitely large). We also observe that the AF phase is more extended in doping in the electron-doped case than in the hole-doped case,13,16 in qualitative agreement with experiments. The results of our calculation also indicate a tendency towards phase separation, which is more strong in the hole-doped case than for electron doping. As we discuss in previous references,13–15,17 this is possibly associated with the larger pseudogap, For a detailed discussion on this issue we refer to these references. >=2

Supercond.

0.12

4 >=3

order param. 0.08 0.04

8 >=4

0 0.8

Antiferrom. order param.

0.6 0.4

10

0.2 0

0.8

0.9

hole

1

n

1.1

1.2

1.3

electron

Fig. 2. AF and SC order parameters as a function of doping for the electron and hole-doped case and for different cluster sizes as specified at the right-hand side of the figure.

3.2. Half metallic ferromagnets: CrO2 In this section, we discuss another class of materials where strong electroncorrelations are important, the so-called half metallic ferromagnets (HMF). These are materials which, according to electronic structure calculations,18 should show

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a metallic behavior for one spin direction (say, spin “up”) and a semiconducting one for the other direction (spin “down”). In other words, the density of states is strongly asymmetric between spin up and spin down, and is gapped around the Fermi energy for spin-down electrons, while for spin up there is a finite density of states. As a consequence, these materials support (ideally) a 100% spin-polarised current at zero temperature and are, therefore, of considerable interest for spin electronics. One should stress that this is an ideal situation which is only valid for bulk materials in the absence of disorder, spin-orbit coupling and correlations. Particularly the latter ones play a crucial role and can change this picture quite drastically. “spin-polaron” processes induce so-called nonquasiparticle states,19,20 i.e. an incoherent density of states in the spin down channel arising within the spin-down gap just above the Fermi energy (see Fig. 3). This is due to the fact that the spin-down low-energy electron excitations, which are forbidden for HMF in the one-particle picture, turn out to be possible as superpositions of spin-up electron excitations and magnons. At zero temperature the density of spin-down states is zero at the Fermi energy. However, with increasing temperature, the Fermi level enters the tail of these nonquasiparticle states, thus quickly reducing the spin polarization below 100% (see Fig. 4). In order to explore correlation effects in specific materials, we have recently carried out a combined VCA+Local Density Approximation (LDA) calculation for CrO2 ,21 further supplemented by a LDA+Dynamical Mean-Field Theory (DMFT) simulation. Based on first-principle calculations, CrO2 is classified as a half-metallic ferromagnet.22 Point contact measurements at superconductor metal interfaces reveal a spin polarization of the conduction electrons larger than 90%,23,24 supporting the half-metallic nature predicted by band theory. In addition, CrO2 fulfils yet another essential requirement for practical purposes, namely a high Curie temperature, determined experimentally in the range of 385–400 K.25 In order to achieve a realistic description of the material, the parameters of the correlated model Hamiltonian are determined via ab-initio electronic-structure calculations. CrO2 has a rutile structure with Cr ions forming a tetragonal bodycenter lattice. Cr4+ has a closed shell Ar core and two additional 3d electrons. The Cr ions are in the center of the CrO6 octahedra. Therefore, the 3d orbitals are split into a t2g triplet and an excited eg doublet. With only two 3d electrons, important correlation effects take place essentially in the t2g orbitals, while eg states can be safely neglected. The cubic symmetry is further reduced to tetragonal due to a distortion of the octahedra, which partially lifts the degeneracy of the t2g orbitals into a dxy ground state and dxz+zy and dxz−zy excited states.26,27 We therefore restrict the VCA calculation to three orbitals on each site representing the Cr-t 2g manifold described by the Hamiltonian 1 X (10) Umm0 m00 m000 c†imσ c†im0 σ0 cim000 σ0 cim00 σ , H = H0 + 2 i{m,σ}

where cimσ destroys an electron with spin σ on orbital m and site i. Here, H0 is

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the noninteracting part of the Hamiltonian restricted to the t2g orbitals obtained by the downfolding procedure implemented within the N-order Muffin-Tin Orbital method28,29 using the LDA. In this way, the effects of the remaining orbitals are included effectively by renormalization of hopping and on-site energy parameters. The on-site Coulomb-interaction terms in Eq. (10) are expressed in terms of the diagonal direct coupling Ummmm = U , the off-diagonal direct coupling Umm0 mm0 = U 0 and the exchange coupling Umm0 m0 m = J.30 Notice that spin-rotation invariance is automatically guaranteed by the form Eq. (10), i.e. spin-flip terms are also included in our calculation. The pair-hopping term is also included via Ummm0 m0 = J. The Coulomb-interaction (U ) and Hund’s exchange (J) parameters between t2g electrons are evaluated from first principles by means of a constrained spin-resolved LDA (LSDA) method,31 giving U ≈ 3 eV and J ≈ 0.9 eV. oxygen p bands

t2g states LSDA

3.0

Spin up

VCA (d orb.)

DOS(states/eV)

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eg states

1.0

0.0 1.0 Spin down

−1.0

−2.0 −8 −7 −6 −5 −4 −3 −2 −1 0

E−EF (eV)

1

2

3

4

5

6

7

nonquasiparticle states

Fig. 3. Spin-resolved density of states for CrO2 obtained by LSDA (dashed line) and by VCA (solid line).

Results for the spin-resolved density of states (summed over the three t2g orbitals) obtained from our VCA calculation are plotted in Fig. 3. For comparison, we also show the results from the LSDA calculation (dashed line). As one can see, LSDA predicts a gap for spin down electrons, with the Fermi level lying within this gap. On the other hand, inclusion of correlations within the VCA approach produce spin down states within the gap just above the Fermi energy, i.e. nonquasiparticle states. These nonquasiparticle states crucially affect the behavior of the spin polarisation as a function of energy, as can be seen in Fig. 4, where we compare the results of different theoretical calculation with results from spin-resolved x-ray absorption experiments.32 The LDA calculation clearly overestimates the spin polarization, as it neglects correlation effects. On the other hand, both VCA and DMFT results show excellent agreement with the experiment at the Fermi level. At an energy of about 0.5 eV, polarization is reduced by about 50%, and up to this energy the

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0.5

0

>=4 −0.5

−1 −0.5

0

0.5

1

1.5

Fig. 4. Energy dependence of the spin polarization obtained experimentally (Ref. 32 circles) and by different theoretical calculations LDA (dashed), DMFT (dotted), VCA (solid). A broadening of 0.4 eV, corresponding to the experimental resolution, has been added to the theoretical curves.

agreement of the VCA calculation is excellent, while DMFT results overestimate depolarisation effects away from the Fermi energy. The disagreement for energies above ≈ 0.5 eV is probably due to terms not included in the VCA Hamiltonian (10), such as eg orbitals which start becoming important at higher energies. 4. Summary In this paper, we have discussed the importance of correlations in two classes of materials, high-temperature superconductors and half-metallic ferromagnets, and presented results of calculations for specific models, carried out within a newly developed cluster method, the VCA. For HTSC, we have presented results for the phase diagram, and discussed the difference between electron- and hole-doped HTSC. For half-metallic ferromagnets, taking CrO2 as an example, we have shown the importance of nonquasiparticle states for a correct description of spin polarization in these materials. Acknowledgments Part of the work presented here originates from very fruitful collaborations with A. Yamasaki, M. Daghofer, M. I. Katsnelson, A. I. Lichtenstein, and M. Potthoff. This work was supported by the Austrian Science Fund (FWF projects P18505-N16 and P18551-N16) and by the Deutsche Forschungsgemeinschaft (FOR538). References 1. 2. 3. 4. 5. 6.

J. G. Bednorz and K. A. M¨ uller, Zeit. Phys. B 64, 189 (1986). N. F. Mott, Proc. Phys. Soc. London, Ser. A 49, p. 72 (1937). M. Imada, A. Fujimori and Y. Tokura, Rev. Mod. Phys. 70, 1039 (1998). L. D. Landau, Sov. Phys. JETP 8, p. 70 (1959). P. W. Anderson, Science 235, p. 1196 (1987). P. Fulde, Electron Correlations in Molecules and Solids (Springer, Berlin, 1995).

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7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

18. 19. 20. 21. 22. 23.

24. 25. 26. 27. 28. 29. 30. 31. 32.

C. Gros and R. Valenti, Phys. Rev. B 48, 418 (1993). D. Sénéchal, D. Perez and M. Pioro-Ladriere, Phys. Rev. Lett. 84, 522 (2000). M. Potthoff, M. Aichhorn and C. Dahnken, Phys. Rev. Lett. 91, p. 206402 (2003). C. Dahnken, M. Aichhorn, W. Hanke, E. Arrigoni and M. Potthoff, Phys. Rev. B 70, p. 245110 (2004). M. Potthoff, Eur. Phys. J. B 32, p. 429 (2003). C. Kim, P. J. White, Z.-X. Shen, T. Tohyama, Y. Shibata, S. Maekawa, B. O. Wells, Y. J. Kim, R. J. Birgeneau and M. A. Kastner, Phys. Rev. Lett. 80, 4245 (1998). M. Aichhorn and E. Arrigoni, Europhys. Lett. 72, 117 (2005). M. Aichhorn, E. Arrigoni, M. Potthoff and W. Hanke, Phys. Rev. B 74, p. 024508 (2006). M. Aichhorn, E. Arrigoni, M. Potthoff and W. Hanke, Phys. Rev. B 74, p. 235117 (2006). D. Sénéchal, P. L. Lavertu, M. A. Marois and A. M. S. Tremblay, Phys. Rev. Lett. 94, p. 156404 (2005). M. Aichhorn, E. Arrigoni, M. Potthoff and W. Hanke, Phase separation and competing superconductivity and magnetism in the two-dimensional hubbard model: From strong to weak coupling, arXiv:0707.3557, (2007). R. A. de Groot, F. M. Mueller, P. G., Engen and K. H. J. Buschow, Phys. Rev. Lett. 50, 2024(Jun 1983). D. M. Edwards and J. A. Hertz, J. Phys F: Met. Phys. 3, 2191(December 1973). V. Y. Irkhin and M. I. Katsnelson, J. Phys.: Condens. Matter 2, 7151 (1990). L. Chioncel, H. Allmaier, A. Yamasaki, M. Daghofer, E. Arrigoni, M. Katsnelson and A. Lichtenstein, Phys. Rev. B 75, p. 140406 (2007). K. Schwarz, J. Phys. F: Met. Phys. 16, L211(September 1986). R. J. Soulen, J. M. Byers, M. S. Osofsky, B. Nadgorny, T. Ambrose, S. F. Cheng, P. R. Broussard, C. Tanaka, J. Nowak, J. S. Moodera, A. Barry and J. M. D. Coey, Science 282, 85(Oct 1998). Y. Ji, G. J. Strijkers, F. Y. Yang, C. L. Chien, J. M. Byers, A. Anguelouch, G. Xiao and A. Gupta, Phys. Rev. Lett. 86, 5585(Jun 2001). S. M. Watts, S. Wirth, S. von Moln´ ar, A. Barry and J. M. D. Coey, Phys. Rev. B 61, 9621(Apr 2000). S. P. Lewis, P. B. Allen and T. Sasaki, Phys. Rev. B 55, 10253(Apr 1997). M. A. Korotin, V. I. Anisimov, D. I. Khomskii and G. A. Sawatzky, Phys. Rev. Lett. 80, 4305(May 1998). O. K. Andersen and T. Saha-Dasgupta, Phys. Rev. B 62, R16219(Dec 2000). A. Yamasaki, L. Chioncel, A. I. Lichtenstein and O. K. Andersen, Phys. Rev. B 74, p. 024419(Jul 2006). A. I. Lichtenstein and M. I. Katsnelson, Phys. Rev. B 57, 6884(Mar 1998). V. I. Anisimov and O. Gunnarsson, Phys. Rev. B 43, 7570(Apr 1991). D. J. Huang, L. H. Tjeng, J. Chen, C. F. Chang, W. P. Wu, S. C. Chung, A. Tanaka, G. Y. Guo, H. J. Lin, S. G. Shyu, C. C. Wu and C. T. Chen, Phys. Rev. B 67, p. 214419 (2003).

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EXCITONS AND POLARITONS IN AN OPTICAL LATTICE FOR COLD-ATOMS WITHIN A CAVITY H. ZOUBI∗ and H. RITSCH Institute for Theoretical Physics, Innsbruck University Technikerstrasse 25, A-6020 Innsbruck, Austria ∗ E-mail: [email protected] We investigate collective electronic excitations for ultracold atoms in an optical lattice prepared in the Mott insulator phase. Frenkel like-excitons in these artificial crystals, similar to Frenkel excitons in Noble atom or molecular crystals, are obtained. They appear when the atomic excited state line width is smaller than the exciton band width. When the optical lattice atoms are placed within a cavity the excitons and the cavity photons get coupled, and in the strong coupling regime they form two branches of cavity polaritons, with Rabi splitting larger than the atomic and the cavity line width. Keywords: Frenkel excitons; cavity polaritons; cold atoms; optical lattices.

A cold dilute gas of boson atoms in an optical lattice has been widely studied both theoretically,1 and experimentally,2 and especially the quantum phase transitions. The optical lattice potential is formed of counter propagating laser beams to get a standing wave with a period of λ/2, half wave length, where the laser is off resonance with any of the atomic transitions. Due to the optical dipole forces, the atoms are confined in an array of microscopic trapping potentials, and in controlling the relevant parameters the optical lattice can be filled with few atoms down to one atom per site. The Bose–Hubbard model can be realized by cold atoms loaded on the optical lattice, and which predicts the phase transition from the superfluid to the Mott insulator phase by changing the optical potential depth.1 In the Mott insulator phase with one atom per site, the cold atoms in an optical lattice can be considered as an artificial crystal, which are similar to molecular crystals, where each atom retains his identity with negligible overlaps between the different site atom electronic wave functions. We study the possibility of the appearance of solid state effects in such artificial crystals. For example in molecular crystals, an electronic excitation localized on a molecule can transfer among the crystal molecules due to electrostatic interactions, without a charge transfer between the molecules. In using the crystal symmetry, such energy transfer can be represented in the quasi-momentum space by a wave that propagates in the crystal, which are called excitons.3 If the molecular crystal is located within a cavity, the excitons and the cavity photons are coupled, and in the strong coupling regime they coherently mix to form new system quasi-

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...

Fig. 1.

ωa

...

J

Energy transfer in 1D atoms chain.

particles which are called cavity-polaritons.3 We investigate here such excitons and polaritons in two-dimensional optical lattice for the case of the Mott insulator with one atom per site.4 We consider two-level atoms, and assume ground and excited optical lattice potentials with minimums at the same positions. The case of two atoms per site is also studied by us.5 The electronic excitations are represented by the Hamiltonian X X Ha = ~ωa Bi† Bi + ~Jij Bi† Bj , (1) i

i,j

where Bi† and Bi are the creation and annihilation operators of an excitation at site i. At low intensity such operators can be taken as Bosons operators.6 The transition frequency ωa includes a light shift relative to free atoms. The energy transfer parameter Jij between sites i and j causes from dipole-dipole interactions, as seen in figure (1). The Hamiltonian can be easily diagonalized by the transformation into P the quasi-momentum space, in using Bi = √1N k Bk eik·ni , where N is the number P of lattice sites. The exciton Hamiltonian reads Ha = k ~ωa (k) Bk† Bk , with the P exciton dispersion ωa (k) = ωa + L J(L) e−ik·L , where we defined the distance between atoms by L = ni − nj . Our system is 2D square with √ a cubic symmetry. The in-plane wave vector is defined by k = (nx , ny ) √ × 2π/ S, where the system area is S = N a2 , and we have (nx , ny = 0, ±1, . . . , ± N/2). If we assume energy transfer only between nearest neighbor sites with coupling parameter J(a) = −J 1 , which is negative and give attractive interactions, hence the exciton dispersion reads ωa (k) = ωa − 2J1 (cos kx a + cos ky a). The energy transfer in a symmetric lattice results in an energy band, with band width of 4~J1 , in place of a discrete level for independent atoms. In fact the dipole-dipole interaction is of long-range interaction, and hence one needs to go beyond the nearest neighbor interactions. But here as the lattice constant is of the order of the atom transition wave length, we limited the discussion for nearest neighbor interactions. In the long wave length limit, that ~k2 , is ka 1 with k = |k|, we get the parabolic dispersion ωa (k) = ωa − 4J1 + 2m ef f 2 where we defined the exciton effective mass by mef f = ~/(2a J1 ). Note that at zero wave vector, k = 0, a significant atomic frequency shift of 4J1 , relative to the free atom frequency, is obtained and which can be easily observed. In order to observe excitons in optical lattice cold atoms, the exciton line width ~Γa needs to be smaller than the exciton band width, namely ~Γa < 4~J1 . The

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Fig. 2.

Optical lattice cold atoms within a cavity.

transfer energy ~J1 is calculated from the dipole-dipole interaction between two ~ atoms of transition dipole µ ~ and which are separated by the distance vector R, 2 and is given by ~J(R) = − 2πµ0 R3 (cos lR + lR sin lR), where ωa = cl, we assumed ~ = RR ˆ z , and for the case of dipoles in the z direction. The distance between two R atoms, for the nearest neighbor coupling, equals the lattice constant. For typical optical lattices the exciton band width is found to be one order of magnitude larger than the free Alkali atom line width and excitons can be observed. We consider now such a quantum gas of ultracold atoms enclosed in the middle of optical resonator built of two parallel mirrors, which we consider as perfect (see Fig. 2). The electromagnetic field is confined in the cavity plane with in-plane wave vector q, and the modes have discrete wave vector kz = mπ/L, where m = 1, 2, . . . in the perpendicular z direction. As the cavity distance in z direction is small we can restrict our considerations to only the m-mode close to resonance with the atomic electronic excitation. As the optical lattice is located in the middle between the cavity mirrors the active mode is taken to be one of the odd modes to ensure strong coupling. For each in-plane wave vector exist two possible polarizations, T E and T M modes. Here we consider only isotropic materials so that we can concentrate in a fixed cavity photon polarization adapted to the atomic excitations. The cavity P Hamiltonian thus is given by Hc = q ~ωc (q) a†q aq , where a†q , aq are the creation and annihilation boson operators of a cavity photon with in-plane wave q vector q. 2 c , The corresponding cavity photon dispersion thus reads ωc (q) = √ q 2 + mπ L where L is the distance between the cavity mirrors, and ≈ 1 is the cavity medium dielectric constant. The excitons are coupled toqthe cavity modes by the dipole 2

c (k)N µ . The coupled excitoninteraction with the coupling parameter ~fk = −i ~ω2LS 0 photon Hamiltonian is written as o X n (2) H= ~ ωa (k) Bk† Bk + ωc (k) a†k ak + fk Bk† ak + fk∗ a†k Bk .

k

where the lattice symmetry ensures that only coupling between excitons and photons with the same in-plane wave vector occurs, i.e. we have quasi-momentum conservation. The above Hamiltonian can be easily diagonalized to give H = P † branches with differkr ~Ωr (k) Akr Akr , which exhibits two diagonal polariton p a (k) 2 2 ± ∆ , where ∆ = ent dispersions Ω± (k) = ωc (k)+ω δ k k k + |fk | and we de2 fined the exciton-photon detuning by δk = (ωc (k) − ωa (k))/2. The splitting be-

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1.5003 1.5002

Upper Polariton Photon Exciton

1.5001 1.5 1.4999

Lower Polariton

1.4998 0

0.5

1 k [Angs.−1]

1.5

2 −5

x 10

Fig. 3. The upper and the lower polariton branches vs. in-plane wave vector k. The dashed line is for the exciton dispersion, and the dashed parabola is for the cavity photon dispersion, using the following numbers: ~ωa (0) = 1.5 eV, L/m = 4133.3 ˚ A, where δ0 = 0, and ~|f | = 0.0001 eV.

tween the two polariton branches at the exciton-photon intersection point, where δk = 0, is 2|fk | which exactly corresponds to the vacuum Rabi splitting. The polaritons thus are coherent superpositions of excitons and photons with oper± ± ators Ak± q = Xk Bk + Yk ak , where the exciton and photon amplitudes are

Xk± = ±

∆k ∓δk 2∆k

and Yk± = √

fk . 2∆k (∆k ∓δk )

At the exciton-photon intersection point

the polaritons are half exciton and half photon, that is |Xk± |2 = |Yk± |2 = 1/2. At large wave vectors the lower branch becomes excitonic, that is |Xk− |2 ≈ 1, |Yk− |2 ≈ 0, and the upper branch becomes photonic, that is |Xk+ |2 ≈ 0, |Yk+ |2 ≈ 1. In Fig. 3 we plot the upper and lower polariton branches. The system presented here will have big applications in optoelectronic devices and quantum information, and can open new directions in optical lattice systems, both theoretically and experimentally. Also the polaritons can be used as a strong observation tool for many physical effects, e.g. optical lattice defects.7 Acknowledgments The work was supported by the Austrian Science Fund (FWF), through the LiseMeitner Program (M977). References 1. 2. 3. 4. 5. 6. 7.

D. Jaksch, et al., Phys. Rev. Lett. 81, p. 3108 (1998). M. Greiner, et al., Nature 415, p. 39 (2002). H. Zoubi, and G. C. La Rocca, Phys. Rev. B 71, p. 235316 (2005). H. Zoubi, and H. Ritsch, Phys. Rev. A 76, p. 13817 (2007). H. Zoubi, and H. Ritsch, arXiv:0707.4432 [quant-ph]. H. Zoubi, and G. C. La Rocca, Phys. Rev. B 72, p. 125306 (2005). H. Zoubi, and H. Ritsch, arXiv:0710.5448 [quant-ph].

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FIXED-NODE QUANTUM MONTE CARLO FOR CHEMISTRY Michel CAFFAREL∗ Laboratoire de Chimie et Physique Quantiques, IRSAMC-CNRS Universit´ e de Toulouse, France ∗ E-mail: [email protected] Alejandro RAMÍREZ-SOLÍS Depto. de Física, Facultad de Ciencias, Universidad Aut´ onoma del Estado de Morelos, Cuernavaca, Morelos 62209, M´ exico E-mail: [email protected] In this paper we discuss the application of quantum Monte Carlo (QMC) techniques to the electronic many-body problem as encountered in computational chemistry. The Fixed-Node Diffusion Monte Carlo (FN-DMC) algorithm —the most common QMC scheme for treating molecules— is presented. The impact of the fixed-node error is illustrated through numerical applications including the calculation of the electronic affinity of the chlorine atom, the dissociation barrier of the O4 molecule, and the binding energy of the dichromium molecule, Cr2 . Although total energies calculated with FN-DMC are very accurate (more accurate than the best alternative methods available), it is emphasized that the error associated with approximate nodes can lead to important errors in the small differences of total energies, quantities which are particularly important in chemistry. Keywords: Quantum Monte Carlo; diffusion Monte Carlo; fixed-node approximation; electronic structure calculations.

1. Introduction At the heart of quantitative chemistry is the formidable mathematical task consisting in finding accurate eigensolutions of the N -body electronic Schr¨ odinger equation. This task is particularly difficult for several reasons. First, the precision required to meet the “chemical” accuracy in realistic applications is very high. For example, in the case of small organic molecules the calculation of atomization energies (the energy needed to break apart a molecule into separated atoms) requires a relative error on the total ground-state energies of at least 10−4 . Calculating electronic affinities, ionization potentials, barriers to dissociation etc. is more demanding and a level of at least 10−5 in accuracy is in general required. For intermolecular forces (hydrogenbonds, van der Waals interaction, etc.) the accuracy needed is at least 10−6 . A second aspect which makes the electronic N -body problem difficult is the fermionic character of electrons. It is well-known that fermions are more difficult to simulate

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than bosons. The basic reason is that fermionic ground-states can be viewed as “mathematically” excited-states of the Hamiltonian and, therefore, display a much more intricate structure in configuration space than bosonic ground-states. Another important aspect concerns the nature of the interaction between electrons and nuclei. The existence of strong attractive nuclei centers localized at fixed positions makes the structure of the electronic distribution highly non-uniform, leading to huge density variations, a situation not easy to describe. Finally, let us emphasize that the electronic structure problem for molecules is a full N -body problem where N is large but finite. Many powerful tools have been developed, in particular in the condensed matter community, to treat efficiently the thermodynamical limit, N → ∞. Here, we face a fully quantitative problem where the small finite variations between the N - and (N + 1)-particle systems are of central importance. To deal with this difficult electronic problem, a number of methods have been developed in the past sixty years. Today, two main approaches are employed. The most popular one is the Density Functional Theory (DFT). In short, DFT approaches are based on the use of approximate energy functionals of the one-body density. A major advantage of DFT is that it is computationally very efficient (computational scaling as N 3 , where N is the number of electrons) and, thus, rather large electronic systems can be considered, up to several thousands of electrons. However, its main weakness is the difficulty in controlling the error made, since the choice of the most appropriate approximate energy functional to deal with a given system is a matter of physical insight and personal experience. The second approach concerns the so-called post-Hartree–Fock methods based on the use of an explicit N -body wavefunction optimized using the variational principle (“abinitio wavefunction-based methods”). There exist many versions of them known under various acronyms like CI (Configuration Interaction), MRCI (MultiReference CI), MPn (perturbational M¨ oller-Plesset of order n), MCSCF (MultiConfiguration Self-Consistent Field), CASSCF (Complete Active Space SCF), CCSD(T) (Coupled Cluster with Single and Double excitations, the Triples treated perturbatively), etc. In contrast with DFT, the accuracy of the ab-initio methods is supposed to be controlled: the larger the size of the one-particle basis set and the greater the order of the method are, the smaller the error is. However, when searching for high accuracy in large fermionic cases, in practice, the exponential increase of the size of the Fock space makes these approaches much less controlled than desired. For a large enough electronic system the size of the tractable basis set is relatively small and the quality of the know-how of the practitioner remains essential to get good results, despite the “ab-initio” character of the approach. The main alternative approaches to these well-established methods are the quantum Monte Carlo (QMC) techniques, a set of stochastic methods to solve the Schr¨ odinger equation. QMC is widely used in the field of quantum solids, quantum liquids, spin systems, and nuclear matter. Successful applications include the uniform electron gas,1 the phase diagrams of hydrogen and helium, the properties of solids,3 etc. In these various fields, QMC is considered as one of the state-of-the-

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art methods for studying complex systems of interacting particles. In contrast, in computational quantum chemistry the situation is different. Despite a number of interesting successes (see, e.g., References in Ref. 4) QMC is still considered as a promising approach but not as an established one. The reasons for that are directly related to the various difficult aspects of the molecular many-body problem listed above, in particular to the problem of reaching a high accuracy in energy differences. In this paper we discuss one of the key difficulties —the fixed-node approximation— encountered when applying QMC to molecules. The organization of this paper is as follows. In the first section, we present a rapid summary of the standard QMC algorithm used for electronic structure calculations, namely the Fixed-Node DMC approach. In the next section, some illustrative examples enlightening the impact of the fixed-node approximation are presented. Finally, we present some remarks summarizing the main points of the paper. 2. Fixed-Node Diffusion Monte Carlo (FN-DMC) 2.1. The DMC algorithm In a quantum Monte Carlo scheme a series of “states”, “configurations”, or “walkers” are generated using some elementary stochastic rules. Here, a configuration is defined as the set of the 3N -electronic coordinates (N number of electrons), the positions of the nuclei being fixed (Born–Oppenheimer approximation) ~ = (r~1 , ..., r~N ). R

(1)

~ may be viewed as a “snapshot” of the molecule Stated differently, a configuration R showing the instantaneous positions of each electron. Stochastic rules are chosen so that configurations are generated according to some target probability density, ~ Note that the probability density is defined over the complete 3N -dimensional Π(R). configuration space and not over the ordinary three-dimensional space. Many variants of QMC can be found in the literature (referred to with various acronyms: VMC, DMC, PDMC, GFMC, etc...). They essentially differ by the type of stochastic rules used and/or by the specific stationary density produced. In the case of the Diffusion Monte Carlo (DMC) approach -the most popular approach for dealing with realistic systems- two basic steps are performed: i.) a standard Monte Carlo step based on the use of a generalized Metropolis algorithm: The walkers are moved using a Langevin-type stochastic differential equation √ ~ new = R ~ old + ~b[R ~ old ]τ + τ η~ , (2) R where τ is an elementary time-step, ~b is the so-called drift vector given by ~ ~b = ∇ψT , ψT

(3)

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where ψT is some approximate trial wavefunction and η~ a Gaussian vector made of 3N independent Gaussian numbers with zero mean and unit variance. ~ new is considered as a “trial” move and is accepted or Each elementary move R rejected according to the acceptance probability q ∈ (0, 1) given by q ≡ M in[1,

~ new )p(R ~ new → R ~ old , τ ) ψT2 (R ], ~ old )p(R ~ old → R ~ new , τ ) ψ 2 (R

(4)

T

~ old → R ~ new , τ ) is the transition probability density corresponding to where p(R Eq. (2), namely ~ old → R ~ new , τ ) = p(R 1 ~ new − R ~ old − ~bold τ )2 ]/2τ }. exp {−[(R (2π)3N/2

(5)

When the move is rejected the new position of the walker is considered to be the old one. It can be easily shown that this generalized Metropolis algorithm admits ψT2 as stationary density (see, e.g. Ref. 5). ii.) a branching (or birth-death) process: Depending on the magnitude of the local energy defined as EL ≡

HψT , ψT

(6)

a given walker can be destroyed (when the local energy is greater than some estimate of the exact energy) or duplicated a certain number of times (local energy lower than the estimate of the exact energy). In practice, the branching step is very easy to implement. After each move the walker is copied a number of times equal to M = Int[exp {−(EL − ET )τ } + u]

(7)

where Int[] is the integer part of a real number, ET some reference energy, and u an uniform random number defined over (0,1). This expression is built so that in average the number of copies is equal to the branching weight exp {−(EL − ET )τ }. Remark that the total number of walkers can now fluctuate and, thus, some sort of population control is required. Indeed, nothing prevents the total walker population from exploding or collapsing entirely. Various solutions to this problem have been proposed. The most popular approaches consist either in performing from time to time a random deletion/duplication step or in varying slowly enough the reference energy, ET , to keep the average number of walkers approximately constant. It can be shown that the stationary density resulting from these rules is given by ~ = ψT (R)φ ~ 0 (R) ~ ΠDM C (R)

(8)

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~ denotes the unknown ground-state wavefunction. It is easy to verify where φ0 (R) that the exact energy is obtained as the average of the local energy over the DMC density: P 1 X ~ (i) ]. E L [R P →+∞ P i=1

E0DM C = lim

(9)

2.2. The Fixed-Node (FN) approximation For bosonic systems the ground-state does not vanish at finite distances and the DMC algorithm just described is exact within statistical uncertainties. In contrast, for fermions the algorithm is slightly biased. To understand this point we first note that the density ΠDM C is necessarily positive by its very definition (as any stationary density resulting from some stochastic rules). As a consequence, φ 0 is not the exact ground-state wavefunction but some approximate one, still solution of the Schr¨ odinger equation but with the additional constraint that φ0 has the same sign as the trial wavefunction everywhere, so that the product in Eq. (8) is always ~ for which the positive. Such a constraint implies that the nodes of φ0 (values of R wavefunction vanishes) are identical to those of the approximate wavefunction ψ T . The resulting error is called the “fixed-node” error. Finally, it can be shown6 that the fixed-node energy is an upper bound of the exact energy, so that FN-DMC is a truly variational method: E0F N −DM C ≥ E0exact .

(10)

The interested reader can find more details about the various QMC algorithms in several excellent reviews, e.g. Refs. 3 or 7. 3. The accuracy of FN-DMC 3.1. Total energies As a general rule, FN-DMC ground-state energies are of a very high quality. For small systems (say, number of electrons smaller than 30) the accuracy achieved is comparable or even superior to that obtained with the best high-level ab-initio methods of computational chemistry. For larger systems, these latter approaches are just not feasible, while QMC calculations are still doable with results of similar quality. The number of electrons that can be treated by QMC is rather large. To the best of our knowledge, the largest FN-DMC all-electron calculation published so far concerns the porphyrine molecule, a system having 182 electrons.8 Using effective core potentials to reproduce the effect of the innermost 1s electron of silicon, Williamson et al.9 were able of computing the ground-state energy of a cluster of silicon and hydrogen atoms containing 986 electrons. To give some illustrative examples we present in Table 1 several calculations performed on various atomic systems of increasing complexity: the chlorine atom (17

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electrons) and its anion (18 electrons), the chromium atom (24 electrons), and the copper atom (29 electrons). All electrons (core and valence) have been included in the simulation and the FN-DMC algorithm presented above has been employed. The trial wavefunction used has a standard form consisting of a Jastrow term multiplied by a one-particle determinantal part, see e.g. Ref. 10. The orbitals have been chosen of the Slater type and are taken from Clementi and Roetti.11 The time step is chosen sufficiently small to get an acceptance rate larger than 0.995. As seen from Table 1 the ground-state energies are very good. Regarding the lighter system, the chlorine atom, the exact non-relativistic energy is known. 12 In quantum chemistry it is usual practice to measure the errors on energies as percents of the so-called correlation energy, defined as the difference of the exact nonrelativistic value and the Hartree–Fock energy. For the chlorine atom the error due to the approximate Hartree–Fock nodes represents only 7% of the correlation energy. Note that this error corresponds to a relative error on the total energy (lowest eigenvalue of the Schr¨ odinger operator) of only 0.1%. In chemistry, such a result is considered as very good. For the Cr and Cu atoms the exact non-relativistic energies are not known. To estimate the accuracy obtained by QMC we have performed for these systems some CCSD(T) calculations using very large atomic basis sets. The CCSD(T) acronym stands for “Coupled Cluster using Single and Double excitations, the Triple excitations being treated in perturbation”.13 When the basis set is large enough and the exact wavefunction is supposed to have a strong monoconfigurational character, CCSD(T) is considered as one of the most accurate methods in computational chemistry. Very large atomic basis sets were used here; for example, in the case of the chromium and copper atoms the primitive basis set are (21s14p7d5f4g3h2i) and (22s17p11d7f5g2h2i), contracted as 244 and 270 one-particle functions, respectively. The number of determinants treated is about 6x108. Although CCSD(T) results are supposed to be very accurate, we see in Table 1 that FN-DMC energies are systematically better(lower). Numerical experience on various molecular systems shows that this result is actually general. The all-electron FN-DMC method can therefore be considered as the method of choice when accurate total ground-state energies of molecular systems are searched for if the constituing atoms do not present important relativistic contributions (Z< 30).

3.2. Difference of energies Getting very accurate total energies is satisfactory, however, it is important to emphasize that the vast majority of chemistry problems does not require precise total energies but, rather, accurate differences of total energies such as barriers, enthalpies, electronic affinities, etc. A general idea valid for any approach is that accurate differences are obtained only when the systematic errors of both components nearly cancel. This is a fundamental point for any ab-initio method and it is,

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359 Table 1. All-electron total ground-state energies at various levels of approximation for Cl, Cl − , Cr, and Cu. Energies in atomic units. Statistical errors on the last digit in parentheses. HF stands for Hartree–Fock, CCSD(T) for Coupled Cluster using Simple and Double excitations, the Triple excitations being treated in perturbation, CE for Correlation Energy. EHF

ECCSD(T )

Cl (Nelec =17) −459.4820 −459.9705 Cl− (Nelec =18) −459.5770 −460.1046 Cr (Nelec =24)) −1043.3559 −1043.8917∗ Cu (Nelec =29)) −1638.9632 −1640.3971 ∗

E0 (Fixed-Node DMC) E0 (exact) CE/% recovered −460.1012(9) −460.2328(24) −1044.310(21) −1640.411(5)

−460.150 −460.283 ? ?

0.668/92.7(1) 0.706/92.9(3) 0.954/? 1.448/?

Averaged Coupled Pair Functional (ACPF) energy for the 7 S ground-state of Cr.

of course, also true for QMC. Within the fixed-node scheme used here it means that the two following conditions need to be fulfilled. First, the statistical fluctuations are to be smaller or much smaller than the difference of energies computed. Second, the fixed-node error on both energy components must almost compensate. Thus, the quality of the results depends very much on the relative magnitudes of these two errors with respect to the desired difference. In Table 2 we present an application of FN-DMC to the computation of the electron affinity (EA) of the chlorine atom (difference of total energies between the neutral atom and its anion, Cl − ). In this example, the energy difference of the order of 10−1 Hartree is large for the standards of most chemical applications. For both atoms the nodes are Hartree–Fock nodes. Note that the Hartree–Fock EA has a large error (about 30%), thus illustrating the importance of electronic correlations. At the FN-DMC level, the statistical error can be easily controlled (about 2%). Within the error bars the FN-DMC result is found to coincide with the experimental result. In this example, the error related to the approximate Hartree–Fock nodes is small. Note also the fact that, in spite that the CCSD(T) energies for Cl and its anion are much less accurate than FN-DMC ones (differences of about 4 eV, see Table 1), their difference is actually very close to the difference of the latter (only 0.08 eV). This latter remark illustrates clearly the very large compensations of errors at work within the ab-initio methods framework. Table 2. Electron affinity (EA) of the chlorine atom in eV. Comparison between Hartree–Fock (STO basis), CCSD(T), Fixed-Node DMC, and exact or experimental results. Statistical errors on the last digit in parentheses.

Electron Affinity

Hartree–Fock

CCSD(T)

FN-DMC

Exact or Expt

2.58

3.65

3.58(7)

3.619,3.62(Expt)

Now, most realistic situations in chemistry involve smaller energy differences. Typical cases concern energy variations of a few kcal/mol, that is ∼ 10 −2 Hartree. At this level of accuracy, numerical experience shows that it is still possible to decrease enough the statistical fluctuations by using properly optimized trial wavefunctions and by performing long enough computations. However, the fixed-node error begins to play a crucial role on these small differences. In Table 3 an illustrative application involving the metastable O4 molecule is presented. The question

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to solve here is to know whether or not the O4 molecule has a sufficiently long life-time to play a role in the dynamical processes at work in the upper atmosphere. Experimental results seem to indicate that it is indeed the case. High-level ab-initio calculations give a slightly too low barrier. Here, the central quantity to determine is the height of the dissociation barrier for O4 into molecular oxygen, that is the difference of total energies between the metastable O4 singlet molecule and the two separated O2 triplet molecules. Although there is no direct experimental value for this barrier, various dynamical models built for polyoxygen species seem to indicate that it should be greater than 10 kcal/mol. The best ab-initio wave-function based approaches lead to a value of about 9.3 kcal/mol, a result that seems to be in contradiction with experimental findings. To elucidate this point, we have performed single reference (Hartree–Fock) and multireference FN-DMC simulations.16 As seen in Table 3 the nature of the nodes plays a crucial role in such calculations. Using Hartree–Fock nodes, the barrier is found to be about 26 kcal/mol. Clearly, such a result is unphysical and is related to the poor quality of the Hartee-Fock nodes. Using a MCSCF (Multi-Configurational-Self-Consistent-Field) trial wave function the barrier is dramatically reduced to a value of 11.4 kcal/mol, in quantitive agreement with the experimental data. Table 3. Dissociation barrier H of the metastable O4 molecule in kcal/mol. Statistical errors on the last digit in parentheses. FN-DMC with HF nodes Barrier a b

H=26.2 ±2.9

FN-DMC with MCSCF nodes

ab-initio

Expt

H= 11.4 ±1.6

H=7.9a ,9.3b

H> 10

CCSD(T)/aug-cc-pVDZ value from14 CASSCF+ACPF/aug-cc-pVQZ benchmark value from15

Another enlightening application showing the impact of the fixed-node approximation in diffusion Monte Carlo is the calculation of the binding energy of the dichromium molecule, Cr2 . One of the most difficult problems in quantum chemistry is to accurately describe the formation and breaking of multiple bonds. In this respect, the electronic structure of the chromium dimer represents a very hard problem to describe, even though it is a “simple” diatomic molecule. The theoretical challenge arises from the fact that the dissociation of the molecular singlet ground-state leads to a couple of separate chromium atoms, each one in their S=7 spin-state (six unpaired electrons in six open shells). Accordingly, Cr2 is considered as a genuine “bête noire” for all high-level ab-initio correlation treatments. In Table 4 we present our all-electron FN-DMC calculations for Cr2 at a bond length of R = 3.2, close to the experimental value. The trial wavefunction used Hartree–Fock nodes and the data are compared to CCSD(T) calculations and experimental ones. A first important remark is that, at the Hartree–Fock level, the Cr2 molecule is not bound by a large amount, +0.795 a.u., although this molecule is experimentally known with a binding energy of about -0.056 a.u. This result illustrates clearly the very strong multiconfigurational character of the exact wavefunction. In quantum

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chemistry it is usual to split the correlation energy (defined above) into two components: a so-called dynamical correlation energy corresponding to the instantaneous effect of the electron-electron interaction and a static or non-dynamical correlation energy associated with a large overlap of the exact wavefunction with many mono-configurational states. Here, the proper description of the S = 0 wavefunction, which is expected to dissociate into two S=7 atomic states, requires to take into account in the zeroth order approximation, at least, all states (determinants) corresponding to the various ways of distributing the twelve valence electrons into the twelve valence orbitals (4s and 3d orbitals for both atoms). The number of such states is about ∼ 800000. In Table 4 we also give the binding energy obtained by Scuseria17 using the CCSD(T) approach. As already mentioned, CCSD(T) is considered as one of the most accurate methods for computing correlation energies. However, since the method is built on a monoconfigurational reference, only the major part of the dynamical correlation energy is recovered. Using a very large basis set, Scuseria found a value of -0.018 kcal/mol. This value is very different from the experimental one and confirms the fact that, in this very complex case, the binding energy is dominated by the non-dynamic correlation energy. We have done extensive FN-DMC calculations using Hartree–Fock nodes. The basis set used is the uncontracted Partridge basis set (20s12p9d).18 We have considered two cases corresponding to using or not f polarization functions (6f basis set functions taken from the Roos Double Zeta ANO basis19 ). As seen from the table, the effect of f functions on the FN-DMC result is important. Note that the importance of f functions in the context of HF theory has already been noticed in a previous work.20 Now, with the largest basis set including f functions we get a FN-DMC binding energy of 0.01(3). Within statistical uncertainty this result is similar to the Scuseria’s result of -0.018. It is interesting to note that, although CCSD(T) and FN-DMC are two completely different methods, similar results are obtained. In the case of the CCSD(T) method, the bad result comes from the mono-configurational character of the reference function on which the coupled cluster ansatz is made. In the DMC case, the origin of the large bias comes from the use of “mono-configurational” nodes.

Table 4. Total energies and binding energy of Cr2 . Bond length R=3.2 Statistical errors on the last digit in parentheses. FN-DMC (HF nodes, no f) FN-DMC (HF nodes, f) Total energies (a.u.) Binding energy (a.u.) a

−2088.522(22)

−2088.612(24)

0.10(3)

0.01(3)

HF

CCSD(T)

Expt.

−2085.917 −2087.516 a 0.795

−0.018

−0.056

1s core electrons not correlated in CCSD(T) (this explains why the CCSD(T) total energy is much higher than the FN-DMC one).

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4. Summary In this paper we have first summarized the various difficult aspects of the manybody problem appearing in quantum chemistry, which make this problem so difficult to solve for atomic and molecular systems. Then, we discussed the present achievements and limitations of quantum Monte Carlo techniques for this problem. After having briefly summarized the main steps of the fixed-node Diffusion Monte Carlo -the standard QMC approach in Chemistry- a number of applications have been presented. It has been illustrated that total ground-state energies obtained with FN-DMC are in general very accurate: the fixed-node error represents only a few percents of the difference between the mean-field result (Hartree–Fock) and the exact energy, the difference being called the correlation energy in quantum chemistry. For small molecules this accuracy is comparable or even superior to that of standard very high-quality ab-initio methods traditionally used in computational chemistry (e.g., Coupled Cluster with large basis sets). However, for larger molecules (let us say, with more than 50 active electrons) Fixed-Node QMC calculations are still feasible with such an accuracy, while other methods are just impossible to implement. However, it has been emphasized that in chemistry total energies are not the most relevant quantities. In general, chemists are much more interested in small differences of energies which define spectroscopic and reactivity quantities like ionization potentials, electroaffinities, reaction enthalpies and activation barriers. For these differences, we have illustrated that the fixed-node approximation can introduce sizable errors. For example, in the case of Cr2 the DMC result using Hartree–Fock nodes predicts an unbound molecule! It is therefore important to construct physically meaningful trial wavefunctions in order to get useful FN-DMC simulations to deal with real chemistry problems. Acknowledgments We wish to thank the UAEM supercomputing center(SEP-FOMES2000), IDRIS (CNRS Orsay), CALMIP (Université de Toulouse) and CNSI (UC Santa Barbara) for unlimited superscalar CPU time. ARS thanks CONACYT(México) through Project No. 45986 and UCMEXUS/CONACYT for sabbatical support at UC Santa Barbara. References 1. D.M. Ceperley and B.J. Alder, Phys. Rev. Lett. 45, 566 (1980). 2. D.M. Ceperley, Rev. Mod. Phys. 67, 279 (1995). 3. W.M.C. Foulkes, L. Mit´ a˘s, R.J. Needs, and G. Rajogopal, Rev. Mod. Phys. 73, 33 (2001). 4. M. Caffarel, J.P. Daudey, J.L. Heully, and A. Ramírez-Solís, J. Chem. Phys. 123, 094102 (2005). 5. M. Caffarel, Introduction to numerical simulations (in french), www.lpthe.jussieu.fr/DEA/

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6. P.J. Reynolds, D.M. Ceperley, B.J. Alder, and W.A. Lester Jr., J. Chem. Phys. 77, 5593 (1982). 7. B.L. Hammond, W.A. Lester Jr., and P.J. Reynolds, Monte Carlo Methods in Ab Initio Quantum Chemistry, World Scientific (1994). 8. A. Aspuru-Guzik, O. El Akramine, J.C. Grossman, and W.A. Lester Jr., J. Chem. Phys. 120, 3049 (2004). 9. A.J. Williamson, R.Q. Hood, and J.C. Grossman, Phys. Rev. Lett. 87, 246406 (2001). 10. R. Assaraf and M. Caffarel, J. Chem. Phys. 113, 4028 (2000). 11. E. Clementi and C. Roetti, Atomic Data and Nuclear Data Tables 14 177 (1974); 12. S.J. Chakravorty, S.R. Gwaltney, and E.R. Davidson, Phys. Rev. A 47, 3649 (1993). 13. e.g., R.J. Bartlett, “Coupled Cluster Theory: an Overview of Recent Developments”, Modern Electronic Structure Theory Part II, World Scientific, Singapore (1995). 14. E.T. Seidl and H.F. Schaefer III, J. Chem. Phys. 96, 1176 (1992). 15. R. Hern´ andez-Lamoneda and A. Ramírez-Solís, J. Chem. Phys. 120, 10084 (2004). 16. M. Caffarel, R. Hern´ andez-Lamoneda, A. Scemama, and A. Ramírez-Solís “On the O4 debate: a multireference quantum Monte Carlo study”, Phys. Rev. Lett (in press). 17. G. E. Scuseria, J. Chem. Phys. 94, 442 (1991). 18. H. Partridge, J. Chem. Phys. 87, 6643 (1987). 19. R. Pou-Amerigo, M. Merch´ an, I. Nebot-Gil, P.O. Widmark and B. Roos, Theor. Chim. Acta 92, 149 (1995). 20. A.D. McLean and B.Liu, Chem. Phys. Lett. 101, 144 (1983).

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QUANTUM MONTE CARLO FOR THE ELECTRONIC STRUCTURE OF ATOMIC SYSTEMS A. SARSA Departamento de F´ısica, Campus de Rabanales, Edif. C2 Universidad de C´ ordoba Cordoba, E-14071 C´ ordoba, Spain E-mail: [email protected] ´ E. BUENDÍA, F. J. GALVEZ and P. MALDONADO Departamento de F´ısica At´ omica Molecular y Nuclear Facultad de Ciencias, Universidad de Granada Granada, E-18071 Granada, Spain In this work we tackle the problem of the electronic structure of atoms by using Quantum Monte Carlo methods. The Variational Monte Carlo method has been extensively employed with trial wave functions which include different correlation mechanisms. A reliable description of different properties such as ionization potentials and electron affinities, or excitation energies is obtained for atoms with a relatively low number of electrons. Variational wave functions are used as starting point in quantum Monte Carlo calculations, Green’s Function and Diffusion Monte Carlo, that provide the exact ground state energy (except for the sign problem). The study of heavier systems is considered. We sketch the extra difficulties from both a computational, due to the high non-homogeneous character of the atomic electron density, and a theoretical point of view. The inclusion of relativistic effects is discussed. Some results for heavier atomic systems are reported. Keywords: Quantum Monte Carlo; Electronic structure of atoms.

1. Introduction Quantum Monte Carlo methods were first developed and applied, about 45 years ago, to nuclear problems.1 It was in 1974 when quantum liquids were tackled by this kind of methods.2 Pioneering studies on the electronic structure of atoms and molecules were carried out in the early eighties.3 Currently, homogeneous systems of nucleons4,5 (neutron and nuclear matter), quantum liquids (4 He and 3 He),6,7 and electron gas,8,9 and finite systems of nucleons10–12 (neutron drops and nuclei), helium drops13–15 (with and without impurities) and electrons16,17 (atoms and molecules) and other many body systems are the focus of extensive research by using different Quantum Monte Carlo methods. The problem of finite temperature systems is also boarded by using the Path Integral Monte Carlo method.18,19 Here we focus on the study of the electronic structure of the atoms. We give a short review of some of the results obtained for this problem, we shall show some

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recent results obtained in our group and outline some of the current problems and perspectives in this field. In order to get accurate results for any many body problem with Quantum Monte Carlo methods, it becomes very important to start from a good wave function of the system. One of the most common correlated wave functions is written as

Ψt = F Φ

(1)

where F is a generalized Jastrow factor and Φ is the model function, a Slater determinant or a linear combination of them, which provides the proper values of the angular momentum and parity of the state under description. For atoms with Z ≤ 10, the model wave function has been usually taken as the Hartree–Fock (HF) solution of the corresponding system, obtaining accurate results.20–23 By using a more sophisticated model wave functions including back-flow type correlations and a multi-configuration model function, high quality results have been recently obtained.24 It is worth to mention here that for these systems, very precise values of the energy based in current Quantum Chemistry methodologies, such as Configuration Interaction,25–27 Coupled Cluster,28–31 Multi Configuration Hartree–Fock,32–34 Perturbation theory35,36 or r12 - Configuration Interaction37,38 methods are available along with very precise semi-phenomenological estimations of the non relativistic ground state energy.39 Much less work can be found in the literature for positive and negative ions,40,41 excited states42,43 or heavier atoms.44,45 From an experimental point of view, excitation energies, electron affinities and ionization potentials have been measured very accurately.46,47 The aim of this work is to show an overview on the different Quantum Monte Carlo methodologies used to describe the low energy spectrum of light and medium atoms starting from simple and accurate wave functions. The structure of this work is as follows. In the next section we present the methodology employed, giving a review of the Monte Carlo methods used to solve the Schr¨ odinger equation and the description of the relativistic effects. In Sec. 3, we show and discuss the results of the present study. Conclusions and perspectives can be found in Sec. 4.

2. Methodology We will discuss here different types of Monte Carlo calculations for atoms, first the Variational Monte Carlo (VMC) method and second the two quantum Monte Carlo (QMC) methods employed, the Diffusion Monte Carlo (DMC) and the Green’s Function Monte Carlo (GFMC). Here we give a brief overview; for a deeper presentation see for example the text of Hammond, Lester and Reynolds.48 It is worth mentioning here that these are not the only quantum Monte Carlo methods for the ground state of a many body systems.49,50

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2.1. Variational Monte Carlo The variational Monte Carlo consists in using the variational method with some approximate trial wave function, along with the Monte Carlo quadrature to calculate the expectation value of the Hamiltonian. Monte Carlo quadratures are based on sampling a large set or points of the configuration space according to a given probability distribution function. Then, the rest of the integrand (not included in the probability distribution function) is averaged in those points giving an estimation of the value of the integral, while the variance gives the numerical uncertainty. In general, the Metropolis algorithm, that allows for the simulation of any given probability distribution function and does not requires knowing the normalization, is used to generate the points of the configuration space. From this point of view, Monte Carlo becomes a numerical quadrature methodology with the fastest convergence features for integration problems of medium and high dimensionality (beyond 3 or 4 dimensions). The importance of this approximation is twofold. First, to have available an approximate wave function is interesting by itself because it provides a reasonable description of the atom, including some dynamic effects beyond the HF approximation (the correlation effects) allowing for the calculation of other properties than the energy that leads to a better understanding of the electronic density of the atom. Second, in the application of Quantum Monte Carlo methods to atoms, it is very convenient to dispose of compact and accurate approximate wave functions, called guiding functions in this framework. Accurate guiding functions give rise to a lowering in the numerical uncertainty. In addition, they play an important role in the so called sign-problem that limits the accuracy (in the sense of convergence to the exact energy) of the QMC methods. Finally, for other properties than the energy, good guiding functions are helpful in the calculation. In the problem of the electronic structure of atoms the trial wave function (1) is widely used. It has been shown that in order to get a reliable description of the atom, it is very important to include not only electron-electron correlations but also electron-nucleus ones. Therefore the correlation Jastrow factor, F , includes electron-electron, electron-nucleus and electron-electron-nucleus correlations. The model function, Φ, is fixed under some approximation that takes into account, in an averaged manner, the electron-electron interaction. This model wave function can be as simple as the Hartree–Fock solution or can be taken as a multi-configuration model function.24,42,51,52 Another alternative is not to use Slater determinants but antisymmetric model functions built from pairing functions.53 A parameterization of the correlation factor that leads to accurate results not only for light atoms20 but also for medium atoms and cations17,54 is P

Uij

(2)

ok ; rij ck (¯ rimk r¯jnk + r¯ink r¯jmk )¯

(3)

F =e

i<j

where Uij =

Nc X

k=1

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and r¯i =

b ri , 1 + b ri

r¯ij =

d rij 1 + d rij

The functional form is based on a averaged back-flow type correlation and presents the correct long and short range behaviors. It is worth mentioning here that the derivatives of the total energy with respect to this parameters is simple allowing for the use of a Newton method or similar to optimize the parameters. By using this form one can impose the electron-electron cusp condition easily. This is an analytic property of the exact wave function that leads to a faster convergence of approximate wave functions that fulfill this condition and reduces the statistical error of Monte Carlo simulations because it removes some divergences of the local energy. 2.2. Quantum Monte Carlo Quantum Monte Carlo method is a technique that allows to simulate the Green’s Function of the ground state of a non-relativistic quantum system of mutually interacting particles. The Green’s Function is built by a large set of random walks. The dynamic associated to the random walks is governed by some probability distribution functions and a transition probability as in the Markov chain. The fixed node3 is an approximation based on using the nodal surface of a variational wave function. This is a widely used approximation that will be employed in the present work. In general, with the fixed node approximation, an upper bound to the exact energy is obtained, while with the exact nodal surface fixed node provides the exact energy. Hence, the more accurate is the trial wave function, the better will be the upper bound to the energy. Therefore trial wave functions including as much as possible exactly known features of the state under study would lead to a better approximation to the nodal surface and therefore to a better bound to the energy. In the atomic problem with the ansatz given in Eq. (1), the nodes can be modified only through the model function Φ. Several forms of this part of the wave function, beyond the HF have been proposed such as a configuration expansion23 (with coefficients fixed at the correlated level), the use of orbitals with back-flow 52 and the use of pairing functions.53 Several algorithms have been proposed to sample the Green’s function of the system.2,3,49,50 Here we shall use two of them, the Diffusion Monte Carlo3 and the Green’s Function Monte Carlo.2 Both lead to the same results if used with the same trial function in a fixed node calculation. In the DMC method a short time approximation is invoked to build and simulate the imaginary time Green’s function. A time step is introduced to perform the simulations. In the limit of zero time step, the exact Green’s function is recovered. In practice, several runs for different time steps are performed and the results are extrapolated to zero time step. The values of the time step required to carry out the extrapolation decrease as the number of electrons in the atom increases.

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In the GFMC method the exact Green’s function of the system is expanded in a Born series by using an auxiliary Green’s function. Monte Carlo is used to sum the series exactly, within the statistical error. This is done by sampling each term of the series with a probability given by its relative weight. Once a term is selected, its contribution is calculated by performing the corresponding Monte Carlo simulation. This procedure is repeated many times to reduce the statistical error. The advantage of this method is that the algorithm is exact, so no extrapolation is required. The major drawback is that the time required for a given simulation with a given error is much larger than several DMC simulations with the same error. In practical implementations of both DMC and GFMC the importance sampling technique2,3 must be used. The idea is to favor those regions of the configuration space where the wave function, at least approximately, takes higher values. This method involves minor changes in the algorithm and gives rise to a significant reduction of the statistical error. The implementation of this technique is based on the knowledge of an approximate wave function of the system. Perfect importance sampling is achieved with the exact eigenfunction, therefore the better the trial wave function the smaller the numerical error for a given length of the run. It is worth to remark here that, although approximate trial functions are used, the calculated energy is exact (with the sign error). Very complex guiding functions (for example very large expansions) can lead to a significant increasing of the computing time per step. Therefore compact and still accurate guiding functions are optimal for using as importance sampling. Finally it is worth to remark here that a quantum Monte Carlo simulation provides the exact value, except for the sign problem, of the ground state energy of the atomic state. This is also the case of the expectation value of those observable that commute with the Hamiltonian. However, there are some interesting properties, such as the radial moments of the single particle density, given by operators that do not commute with H. For these quantities, the extrapolation estimator provides an approximation to the exact value that it is obtained straightforwardly from the value obtained out of a quantum Monte Carlo calculation and the corresponding VMC one. This estimator is not exact and depends on the quality of the guiding function. Several algorithms have been developed in order to obtain the exact value of these observable.22,55–58 In general, they can be incorporated into the algorithm without further increase of the computing time. A different strategy is to use the Reptation Quantum Monte Carlo49 or the Path Integral Ground State50 methods that sample, in a different manner, the time dependent propagator and provide unbiased estimators for these operators.

2.3. Relativistic effects The relativistic problem is solved here within the numerical-parameterized relativistic optimized effective potential (NPROEP) method.59 This is a relativistic version of the numerical-parameterized optimized effective potential (NPOEP) approxima-

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tion.60 Both constitute and alternative to the Hartree–Fock and Dirac–Hartree–Fock (DHF) methods. The method is variational with a trial function of the same form as in the HF or DHF methods. However, the orbitals in the Slater determinants are eigenfunctions of a single particle Schr¨ odinger equation (non-relativistic) or Dirac equation (relativistic) with a single particle potential, called the effective potential. Once the orbitals are fixed, the Slater determinants are build and the expectation value of the N -electron Hamiltonian is computed. Thus the total energy of the system becomes a functional of the effective potential that provides the orbitals. The optimized potential is obtained by minimizing the total energy of the atom with respect to the effective potential. Finally, numerical-parameterized stand for the numerical solution of the single particle Schr¨ odinger or Dirac equation and the parameterized form employed for the effective potential.60 This method approaches very well the Hartree–Fock results (with relative differences below 0.01%), simplifies the technical problem, especially for excited states, and can be straightforwardly extended to the multi-configuration case or to a relativistic Hamiltonian as it is done here. The N -electron relativistic Hamiltonian employed in this work is X X X 1 + VB (i, j) (4) H= hD (i) + r i<j i i<j ij where hD (i) is the single-particle Dirac Hamiltonian for the i-th electron hD (i) = c~ αi · p~i + c2 βi + Vn (r) where Vn (r) is the nuclear potential, taken as the electrostatic potential due to a spherical uniform charge distribution of radius R, and VB (i, j) is the Breit operator, which in the ωij rij /c 1 approximation, where ωij is the energy of the exchanged photon between the two electrons, is given by61 ( ) α~i · α~j (α~i · ~ri )(α~j · ~rj ) VB (i, j) = − + . (5) 3 rij rij 2.4. Importance of correlations and relativistic effects From a quantitative point of view, correlation effects are more important for atoms with 2 ≤ Z ≤ 12, whereas the relativistic correction is higher than the correlation one for heavier atoms. In the left hand side panel of Fig. 1 we show both corrections to the POEP energy for atoms 2 ≤ Z ≤ 18 as a function of the nuclear charge. Relativistic energy grows much faster than correlation energy as can be seen from the figure. However, much of the properties of interest, such as the ionization potential or excitation energies involve energy differences. For the ionization potential, a great amount of the relativistic correction of the cation and neutral atom cancel out. In the right hand side panel of Fig. 1 we show the relativistic correction to the ionization potential as compared to the correlation correction to this quantity for atoms 2 ≤ Z ≤ 18. Both corrections are with respect to the NPOEP values.

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Correction to NPOEP energy

Correction to the Ionization Potential

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Relativistic Correlaton 0.06 0.05

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Delta IP (h)

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Delta E (h)

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Fig. 1. Left: Comparison of the relativistic and correlation energies (in different approximations), in hartree for atoms. Right: Comparison of the relativistic and exact correlation corrections to the ionization potential, in hartree, for atoms with 2 ≤ Z ≤ 18.

3. Results In Table 1 we report the VMC and DMC energies for the Mg, Al and Si atoms calculated from different trial wave functions. As one can see, the use of a wave Table 1. Ground state energy in atomic units for the Mg, Al and Si atoms from different approximations. SC and CI(2) stand for a single and two configurations model wave function respectively. In square brackets we report the percentage of correlation energy.

Mg

Al

Si

HF62 −199.614636 MR-SDCI27 −200.02520[94] HF62 −241.876707 MR-SDCI27 −242.31673[94] HF62 −288.854363 MR-SDCI27 −289.31971[92]

VMC-SC17 −199.9865(5)[85] DMC-SC −200.0340(7)[96] VMC-SC17 −242.2685(5)[84] DMC-SC −242.3200(7)[95] VMC-SC17 −289.2697(5)[82] DMC-SC −289.326(3)[93]

VMC53 −200.0002(5)[88] DMC53 −200.0389(5)[97] VMC53 −242.2124(9)[72] DMC53 −242.3265(10)[96] VMC53 −289.197(1)[68] DMC53 −289.3285(10)[94]

VMC-CI(2)54 −200.0002(4)[88] DMC-CI(2) −200.0390(6)[97] VMC-CI(2)54 −242.2751(5)[85] DMC-CI(2) −242.3250(7)[96] VMC-CI(2)54 −289.2732(5)[83] DMC-CI(2) −289.329(2)[94]

exact39 −200.053 exact39 −242.346 exact39 −289.359

function including two configurations leads to a lowering of the ground state energy for both VMC and DMC. The quantitative value of the correlation energy recovered is similar to that obtained for the corresponding first row atoms of the same group. 23 In figure 2 we plot for 1 ≤ Z ≤ 18 atoms the ionization potential and the electron affinity in the left and right hand side panels respectively. As it is clear from Fig. 2, the inclusion of correlations is very important in order to achieve a better quantitative agreement with the experimental results. The improvement induced by relativity is in the right direction but, in most of the cases, is not enough to reach the experimental values. The trend of both, correlations effect and relativistic correction is to improve the non correlated non relativistic values.

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Atomic ionization potential

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5 NPOEP Rel Correl Exp

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Fig. 2. Ionization potential, left hand side panel, and Electron Affinity, right hand side panel in eV for 1 ≤ Z ≤ 18 calculated from different approximations as compared with the experimental values

In Table 2 the results for the ground state of the Fe atom, as well the ionization potential and the excitation energy of the the 1 S excited state, which lies experimentally in the continuum, are reported. The inclusion of correlations and Table 2. For the Fe atom, ground state energy in atomic units, ionization potential in eV and excitation energy in eV of the [Ar]3d6 4s2 1 S excited state. Experimentally this state does not lie in the discrete spectrum. E IP(eV) EA(eV) ∆E(1 S)(eV)

NPOEP −1262.42539 6.4372 −2.4191 4.9414

RNPOEP −1271.52694 6.7172 −2.5796

VMC63 −1263.20(2)

VMC54 −1263.376(2) 7.51(8) −0.11(8) 4.7(1)

GFMC −1263.550(4) 7.6(2) 6.5(5)

R-VMC

R-GFMC

Exp

7.88(8) −0.27(8)

7.9(2)

7.9024 0.151(3)

relativity improve substantially the non correlated non relativistic values. Although the statistical error is still big, one can say that the Green’s Function Monte Carlo ionization potential improves the Variational value for this quantity. Although correlations improve the non-correlated electron affinity, the result is still far from the experimental value, in fact it has opposite sing. For this property, relativistic effects contribute in the wrong direction. A more accurate wave function for the anion, for example including more configurations, is required to approach the experimental value. With respect to the [Ar]3d6 4s2 1 S excited state of the Fe atom, both HF and VMC does not provide an excitation energy within the continuum spectrum. A GFMC calculation provides higher value of the excitation energy, closer to the ionization energy of the Fe atom. Therefore, although quantitatively GFMC does not give the correct value, the correction to both non correlated and Variational energy is in the right direction. 4. Conclusion The effect of correlations and relativity on the electronic structure of atoms have been discussed. Correlations are studied by using Quantum Monte Carlo method,

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while relativistic effects have been considered in a non correlated framework. If these two dynamical effects are considered a better agreement with the experimental values for both the ionization energy and the electron affinity is obtained for the atoms of the first and second row. The ground state energy and the ionization potential of the Fe atom have been calculated by using VMC and GFMC, and relativistic corrections have been added perturbatively. The excitation energy of the 1 S state of this atom lying in the continuum experimentally have been calculated by using these methods. The GFMC improves the variational result, although gives an excitation energy smaller than the experimental ionization potential. References 1. M. H. Kalos, Phys. Rev. 128, 1791 (1962). 2. M. H. Kalos, D. Levesque and L. Verlet, Phys. Rev. A 9, 2178 (1974). 3. P. J. Reynolds, D. Ceperley, B. J. Alder and W. A. Lester Jr., J. Chem. Phys. 77, 5593 (1982). 4. S. Fantoni, A. Sarsa and K. E. Schmidt, Phys. Rev. Lett 87, 181101 (2001). 5. S. Gandolfi, F. Pederiva and S. F. K. E. Schmidt, Phys. Rev. Lett 98, 102503 (2007). 6. J. Casulleras and J. Boronat, Phys. Rev. Lett 84, 3121 (2000). 7. A. Sarsa, J. Mur-Petit, A. Polls and J. Navarro, Phys. Rev. B 68, 224514 (2003). 8. M. Holzmann, D. M. Ceperley, C. Pierleoni and K. Esler, Phys. Rev. E 68, 046707 (2003). 9. S. D. Palo, M. Botti, S. Moroni and G. Senatore, Phys. Rev. Lett. 94, 226405 (2005). 10. S. C. Pieper, K. Varga and R. B. Wiringa, Phys. Rev. C. 66, 044310 (2002). 11. F. Pederiva, A. Sarsa, K. E. Schmidt and S. Fantoni, Nucl. Phys. A 742, 255 (2004). 12. E. Buend´ıa, F. J. G´ alvez and A. Sarsa, Phys. Rev. C 70, 054315 (2004). 13. A. Sarsa, Z. Baˇcić, J. W. Moskowitz and K. E. Schmidt, Phys. Rev. Lett 88, 123401 (2002). 14. R. Guardiola and J. Navarro, Phys. Rev. A 71, 035201 (2005). 15. E. Sola, J. Casulleras and J. Boronat, Phys. Rev. B 73, 092515 (2006). 16. A. Ma, N. Drummond, M. D. Towler and R. J. Needs, Phys. Rev. E 71, 066704 (2005). 17. E. Buend´ıa, F. J. G´ alvez and A. Sarsa, Chem. Phys. Lett. 428, 241 (2006). 18. D. M. Ceperley, Rev. Mod. Phys. 67, 279 (1995). 19. S. Pilati, K. Sakkos, J. Boronat, J. Casulleras and S. Giorgini, Phys. Rev. A 74, 043621 (2006). 20. K. E. Schmidt and J. W. Moskowitz, J. Chem. Phys. 93, 4172 (1990). 21. A. L¨ uchow and J. B. Anderson, J. Chem. Phys. 105, 7073 (1996). 22. P. Langfelder, S. M. Rothstein and J. Vrbik, J. Chem. Phys 107, 8525 (1997). 23. F. J. G´ alvez, E. Buend´ıa and A. Sarsa, J. Chem. Phys. 115, 1166 (2001). 24. M. D. Brown, J. R. Trail, P. L. R´ıos and R. J. Needs, J. Chem. Phys. 126, 224110 (2007). 25. F. Sasaki and M. Yoshimine, Phys. Rev. A 9, 17 (1974). 26. F. Sasaki and M. Yoshimine, Phys. Rev. A 9, 26 (1974). 27. H. Meyer, T. M¨ uller and A. Schweig, Chem. Phys. 191, 213 (1995). 28. L. Adamowicz and R. J. Barlett, Phys. Rev. A 37, 1 (1986). 29. G. E. Scuseria, J. Chem. Phys. 95, 7426 (1991). 30. C. E. Campbell, E. Krotscheck and T. Pang, Phys. Rep. 213, 1 (1992). 31. R. Bukowski, B. Jeziorski and K. Szalewicz, J. Chem. Phys. 110, 4165 (1999). 32. C. F. Fischer, J. Phys. B 26, 855 (1993).

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33. J. Carlsson, P. J¨ onsson, M. R. Godefroid and C. F. Fischer, J. Phys. B 28, 3729 (1995). 34. M. R. Godefroid, G. van Meulebeke, P. J¨ onsson and C. F. Fischer, Z. Phys. D 42, 193 (1997). 35. Z. J. Wu and Y. Kawazoe, Chem. Phys. Lett 423, 81 (2006). 36. I. Lindgren, Phys. Script. T120, 15 (2005). 37. G. Buesse, H. Kleindienst and A. L¨ uchow, Int. J. Quantum Chem. 66, 241 (1998). 38. R. J. Gdanitz, J. Chem. Phys. 109, 9795 (1998). 39. S. J. Chakravorty, S. R. Gwaltney, E. R. Davidson, F. Parpia and C. F. Fischer, Phys. Rev. A 47, 3649 (1993). 40. J. W. Moskowitz and K. E. Schmidt, J. Chem. Phys. 97, 3382 (1992). 41. F. J. G´ alvez, E. Buend´ıa and A. Sarsa, Int. J. Quantum Chem. 87, 270 (2002). 42. F. J. G´ alvez, E. Buend´ıa and A. Sarsa, J. Chem. Phys. 124, 044319 (2006). 43. T. Oyamada, K. Hongo, Y. Kawazoe and H. Yasuhara, J. Chem. Phys. 125, 014101 (2006). 44. M. Caffarel, J. P. Daudey, J. L. Heully and A. Ram´ırez-Solis, J. Chem. Phys. 123, 094102 (2005). 45. E. Buend´ıa, F. J. G´ alvez and A. Sarsa, J. Chem. Phys. 124, 154101 (2006). 46. T. Andersen, H. K. Hauge and H. Hotop, J. Phys. Chem. Ref. Data 28, 1511 (1999). 47. http://physics.nist.gov/physrefdata/asd/index.html. 48. B. L. Hammond, W. A. Lester Jr. and P. J. Reynolds, Monte Carlo Methods in ab initio Quantum Chemistry (World Scientific, Singapore, 1994). 49. S. Baroni and S. Moroni, Phys. Rev. Lett 82, 4745 (1999). 50. A. Sarsa, K. E. Schmidt and W. R. Magro, J. Chem. Phys. 113, 1366 (2000). 51. H. J. Flad, M. Caffarel and A. Savin, Quantum monte carlo calculations with multireference trial wave functions, in Recent advances in quantum monte carlo methods, ed. W. A. Lester Jr. (World Scientific, Singapore, 1997) 52. N. Drummond, P. L. R´ıos, A. Ma, J. Trail, G. Spink, M. D. Towler and R. J. Needs, J. Chem. Phys. 124, 224104 (2006). 53. M. Casula and S. Sorella, J. Chem. Phys. 119, 6500 (2003). 54. E. Buend´ıa, F. J. G´ alvez and A. Sarsa, Chem. Phys. Lett. 436, 352 (2007). 55. S. Zhang and M. H. Kalos, J. Stat. Phys. 70, 515 (1993). 56. J. Casulleras and J. Boronat, Phys. Rev. B 52, 3654 (1995). 57. A. Sarsa, J. Boronat and J. Casulleras, J. Chem. Phys 116, 5956 (2002). 58. M. H. Kalos and F. Arias de Saavedra, J. Chem. Phys. 121, 5143 (2004). 59. E. Buend´ıa, F. J. G´ alvez, P. Maldonado and A. Sarsa, J. Phys. B: At. Mol. Opt. Phys. 40, 3045 (2007). 60. E. Buend´ıa, F. J. G´ alvez, P. Maldonado and A. Sarsa, J. Phys. B: At. Mol. Opt. Phys. 39, 3575 (2006). 61. J. P. Santos, G. C. Rodrigues, J. P. Marques, F. Parente, J. P. Desclaux and P. Indelicato, Eur. Phys. J. 37, 201 (2006). 62. C. F. Bunge, J. A. Barrientos and A. V. Bunge, At. Data and Nucl. Data Tables 53, 113 (1993). 63. W. M. C. Foulkes, L. Mitas, R. L. Needs and G. Rajagopal, Rev. Mod. Phys. 73, 33 (2001).

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HIERARCHICAL METHOD FOR THE DYNAMICS OF METAL CLUSTERS IN CONTACT WITH AN ENVIRONMENT G. BOUSQUET, P. M. DINH, J. MESSUD, and E. SURAUD∗ Laboratoire de Physique Th´ eorique, UMR CNRS – Universit´ e Paul Sabatier, 118, route de Narbonne F-31062 Toulouse C´ edex, France ∗ E-mail: [email protected] M. BAER, F. FEHRER, and P.-G. REINHARD∗∗ Institut f¨ ur Theoretische Physik, Universit¨ at Erlangen, Staudtstrasse 7, D-91058 Erlangen, Germany , ∗∗ E-mail: [email protected] We present a recently introduced hierarchical model for the description of clusters in contact with an environment (embedded or deposited cluster). We briefly outline the ingredients of the model and show the relevance of a proper treatment of the degrees of freedom of the environment, taking as example the optical response of a small embedded metal cluster. We finally discuss the technical and formal difficulties raised by a proper treatment of the Self Interaction Correction problem in a true dynamical situation. Keywords: Metal clusters; rare gas matrix; time-dependent density-functional theory; self-interaction correction.

Nanotechnology was predicted long ago to stimulate an industrial revolution 1 and it now constitutes a fast developing research field, for example in order to increase the control of material at nanometer scale.2 Metal clusters have especially attracted much attention in the past decades,3–7 in particular at the side of their optical properties.3 The Mie surface plasmon is known to play a key role in any dynamical regime,7 as it provides a strong coupling to photons in a very narrow frequency window which makes clusters an ideal laboratory for laser induced non-linear dynamics.8,9 An even richer variety of phenomena emerges when considering metal clusters in contact with other materials (embedded in a matrix or deposited on a surface), both in terms of fundamental questions as in terms of potential applications such as for example the dedicated shaping of clusters with intense laser pulses.10,11 As the environment simplifies the handling many interesting ongoing experiments can only be done with clusters in contact with a carrier material.12–15 Clusters in contact with insulators are also a versatile model system for chromophores which can be used, for studies of radiation damage in materials16,17 or as indicators in biological tissues.18,19 Embedded clusters nevertheless represent a great theoretical challenge because of the necessary handling of the cluster-matrix interface and

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because of the (necessary) huge number of atoms in the matrix. This holds even more so for the simulation of proper dynamical scenarios. But the broad range of applications has motivated the development of a robust and efficient hierarchical scheme for the dynamics of metal clusters in contact with rare gas materials.20–28 This model relies on a mix of quantum-mechanical and classical descriptions. The Na cluster is treated in full microscopic detail. We use quantum-mechanical singleparticle wavefunctions for the valence electron, which are coupled non-adiabatically to classical molecular dynamics (MD) for the Na ions, described by their positions. The electronic wavefunctions are propagated by time-dependent local-density approximation (TDLDA) and accounting for Self Interaction Correction (SIC) effects through the Averaged Density SIC (ADSIC) scheme.29 The electron-ion interaction inside the cluster is described by soft, local pseudo-potentials. The TDLDA-MD model has been widely validated for linear and non-linear dynamics in the case of free metal clusters.7,8 The rare gas environment (Ar in the examples shown below) is treated at a classical level, associating two degrees-of-freedom to each atom, position and electrical dipole moment. The atomic dipoles allow to explicitly treat the dynamical polarizability of the atoms through polarization potentials .30 The Coulomb field of the Ar dipoles provides the polarization potentials which are the dominant agents at long range. The parameters of the Ar atoms are adjusted to reproduce the dynamical polarizability αD (ω) of the Ar atom at low frequencies. The Na+ ions of the metal cluster predominantly interact with the Ar dipoles by their monopole moment. The small dipole polarizability of the Na+ core is neglected. The cluster electrons do also naturally couple to the Coulomb field generated by the atoms (cores and electron clouds). The short-range repulsion is obtained by adding local core potentials: Lennard–Jones type potential for the Ar-Ar core interaction, Na-Ar core potential following.31 The electron-Ar core repulsion has been modeled following.32 Its parameters determine sensitively the binding properties of Na to the Ar atoms. That piece is used to allow for a final fine-tuning of the model, the benchmark being provided by the Na-Ar dimer.33,34 The Van-der-Waals interaction is known to be a crucial detail determining the Na-Ar binding and it is also accounted for in an approximate way.21 The optical response is a well known key quantity for the analysis of cluster structure properties, especially in the case of metal clusters.4,35 It is thus an interesting testing ground for analyzing the soundness of the model presented above. The reliability of the TDLDA treatment has been fully demonstrated in the case of free clusters7,8,36,37 and it thus constitutes a safe starting basis for investigating the impact of the environment. This is illustrated in Fig. 1 in the case of a Na8 cluster embedded inside an Ar “matrix” of 164 Ar atoms. Systematic variations of the number of Ar atoms have shown that 164 represents a reasonable approximation of larger matrices. Indeed, while at small Ar coverages, one observes specific size effects, large numbers of Ar atoms allow to identify systematic trends and behaviors. Beyond 164 atoms, one clearly enters the second class of systems. Three calculations are presented in Fig. 1 : a free cluster and two embedded ones, one with the full

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Optical response of Na8 for three cases, as indicated.

model and one within freezing the Ar dipoles. This shows the importance of this dipolar degree of freedom which half cancels the (short-range) compressing effect (thus blue-shifting the response) of the Ar atoms by adding a (long-range) attraction through polarization effects (resulting a red-shift of the response). The net shift of the dipole is finally small, in qualitative agreement with available experimental data on embedded Ag.15,26 The case of much smaller systems, which do not exhibit per se an optical response is also interesting as it allows a comparison to detailed quantum chemistry calculations. As an example, we consider the case of a single Na atom in contact with a small Ar6 cluster. The analogon of the optical response is here the 3s −→ 3p transition and its splitting which has been widely studied in the past.33,38–40 The average value obtained in38 is 2.12 eV with a splitting of 0.08 eV coming from the splitting of the 3p state in presence of the Ar cluster. We find in our calculations an average value of 2.01 eV (LDA, 2D cylindrically symmetric calculation) with a splitting of 0.07 eV, 2.26 eV (ADSIC, 2D) with splitting 0.05 and 2.23 eV (ADSIC, full 3D calculation with no imposed symmetry) with splitting 0.07. All calculations do very well agree with each other, up to details that we do not have space to discuss here. An interesting aspect, though, is the fact that LDA performs as good as ADSIC what concerns the transition energy, while it leads to very different energies of single particle levels (mind that we only have access to transition energies from, 38 not single particle energies). The Ionization Potential (IP), in particular, differ by almost a factor 2 between LDA and ADSIC and it is well known that the LDA value is usually wrong.29 The LDA might thus suffice for optical response calculations, as in the case of free clusters but it becomes irrelevant when ionization processes are considered. And irradiation is precisely one of the goals of our investigations. We thus have to properly take into account the SIC in dynamics. The ADSIC approach, beyond its simplicity, exhibits several interesting formal properties (unitarity, hermiticity, . . . )29 and has been successfully used in both

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metal clusters29 and organic compounds.41 As such, it can a priori be easily transferred to dynamical situations. However it suffers from a fundamental defect : because of the involved average, it is irrelevant for dissociation,29 a process which we would also like to consider, especially in the case of irradiated organic molecules. This shortcoming has motivated specific studies on an alternative, time dependent robust, formulation of a SIC, not raising this delicate question of dissociation defect. The first investigations led on this question have driven us back to consider a proper reformulation of the original SIC approach,42 trying to improve it, especially in its tim dependent version, what concerns hermiticity and unitarity robustness, a basic defect of this orbital dependent DFT approach. Work is in progress along that line. We have presented in this paper a hierarchical approach for the description of the dynamics of metal clusters in contact with an environment. The term hierarchical refers to the fully microscopic description of the cluster, while the environment is treated at a simpler level of sophistication, mostly in terms of a classical polarizable medium. We have chosen here as test case an environment constituted by Ar atoms (embedded or deposited cluster) as a prototype of moderately interacting environment. However we have also led calculations on other rare gases26 and on an ionic insulator, MgO.43 In all cases, as illustrated in the present contribution, we have found that, even in the case of relatively inert environments, a proper and detailed treatment of the degrees of freedom of the latter is crucial. The detailed description of the cluster electrons is also decisive and we have pointed out the basic difficulty with a proper treatment of the Self Interaction Correction in truly dynamical processes, involving electronic excitations, as the ones we aim at describing with the kind of hierarchical model we have presented here. Work to improve our approaches in these many respects is in progress. Acknowledgments This work was supported by the DFG, project nr. RE 322/10-1, the French-German exchange program PROCOPE nr. 04670PG, the CNRS Programme “Matériaux” (CPR-ISMIR), Institut Universitaire de France, the Humbodlt foundation and a Gay-Lussac price. References 1. R. Feynmann, Engineering and Science 23, p. 22 (1960). 2. W. Eberhardt, Surf. Sci. 500, p. 242 (2002). 3. U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters (Springer Series in Materials Science, 1993). 4. W. A. de Heer, Rev. Mod. Phys. 65, p. 611 (1993). 5. H. Haberland (ed.), Clusters of Atoms and Molecules 1 and 2 (Springer Series in Chemical Physics, Berlin, 1994). 6. W. Ekardt (ed.), Metal Clusters (Wiley, New York, 1999). 7. P.-G. Reinhard and E. Suraud, Introduction to Cluster Dynamics (Wiley, New York, 2003). 8. F. Calvayrac, P.-G. Reinhard, E. Suraud and C. A. Ullrich, Phys. Rep. 337, p. 493 (2000).

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9. S. Teuber, T. D¨ oppner, T. Fennel, J. Tiggesb¨ aumker and K. H. Meiwes-Broer, Euro. Phys. J. D 16, p. 59 (2001). 10. G. Seifert, M. Kaempfe, K.-J. Berg and H. Graener, Appl. Phys. B 71, p. 795 (2000). 11. H. Ouacha, C. Hendrich, F. Hubenthal and F. Tr¨ ager, Appl. Phys. B 81, p. 663 (2005). 12. N. Nilius, N. Ernst and H.-J. Freund, Phys. Rev. Lett. 84, p. 3994 (2000). 13. J. Lehmann, M. Merschdorf, W. Pfeiffer, A. Thon, S. Voll and G. Gerber, Phys. Rev. Lett. 85, p. 2921 (2000). 14. M. Gaudry, J. Lermé, E. Cottancin, M. Pellarin, J.-L. Vialle, M. Broyer, B. Prével, M. Treilleux and P. Mélinon, Phys. Rev. B 64, p. 085407 (2001). 15. T. Diederich, J. Tiggesb¨ aumker and K. H. Meiwes-Broer, J. Chem. Phys. 116, p. 3263 (2002). 16. M. Bargheer, M. Guhr and N. Schwentner, J. Chem. Phys. 117, p. 5 (2002). 17. M. Y. Niv, M. Bargheer and R. B. Gerber, J. Chem. Phys. 113, p. 6660 (2000). 18. C. Mayer, R. Palkovits, G. Bauer and T. Schalkhammer, J. Nanoparticle Res. 3, p. 361 (2001). 19. B. Dubertret, P. Skourides, D. J. Norris, V. Noireaux, A. H. Brivanlou and A. Libchaber, Science 298, p. 1759 (2002). 20. B. Gervais, E. Giglio, E. Jaquet, A. Ipatov, P.-G. Reinhard and E. Suraud, J. Chem. Phys. 121, p. 8466 (2004). 21. F. Fehrer, P.-G. Reinhard, E. Suraud, E. Giglio, B. Gervais and A. Ipatov, Appl. Phys. A 82, p. 151 (2005). 22. J. Douady, B. Gervais, E. Giglio, A. Ipatov and E. Jacquet, J. Mol. Struct. 786, p. 118 (2006). 23. P. M. Dinh, F. Fehrer, P.-G. Reinhard and E. Suraud, Intern. J. Quant. Chem. (2007, in press). 24. P. M. Dinh, F. Fehrer, P.-G. Reinhard and E. Suraud, Euro. Phys. J. D online first (2007). 25. P. M. Dinh, F. Fehrer, G. Bousquet, P.-G. Reinhard and E. Suraud, Phys. Rev. A (2007, in press). 26. F. Fehrer, P. M. Dinh, P-G, Reinhard and E. Suraud, Phys. Rev. B 75, p. 235418 (2007). 27. F. Fehrer, P. M. Dinh, M. Baer, P-G, Reinhard and E. Suraud, Euro. Phys. J. D online first (2007). 28. F. Fehrer, P. M. Dinh, P-G, Reinhard and E. Suraud, Comp. Mat. Sci. (2007, in press). 29. C. Legrand, E. Suraud and P.-G. Reinhard, J. Phys. B 35, p. 1115 (2002). 30. B. G. Dick and A. W. Overhauser, Phys. Rev. 112, p. 90 (1958). 31. G. R. Ahmadi, J. Alml¨ of and J. Roegen, Chem. Phys. 199, p. 33 (1995). 32. F. Dupl` ae and F. Spiegelmann, J. Chem. Phys. 105, p. 1492 (1996). 33. M. Gross and F. Spiegelmann, J. Chem. Phys. 108, p. 4148 (1998). 34. M. B. E. H. Rhouma, H. Berriche, Z. B. Lakhdar and F. Spiegelman, J. Chem. Phys. 116, p. 1839 (2002). 35. M. Brack, Rev. Mod. Phys. 65, p. 677 (1993). 36. K. Yabana and G. F. Bertsch, Phys. Rev. B 54, p. 4484 (1996). 37. K. Yabana and G. F. Bertsch, Phys. Rev. A 58, p. 2604 (1998). 38. C. Tsoo, D. A. Estrin and S. J. Singer, J. Chem. Phys. 96, p. 7997 (1992). 39. J. A. Boatz and M. E. Fajardo, J. Chem. Phys. 101, p. 3472 (1994). 40. M. E. Fajardo, P. G. Carrick and J. W. Kenney, J. Chem. Phys. 94, p. 5812 (1991). 41. I. Ciofini, C. Adamo and H. Chermette, Chem. Phys. Lett. 380, p. 12 (2003). 42. J. P. Perdew and A. Zunger, Phys. Rev. B 23, p. 5048 (1981). 43. D. of metal clusters in rare gas clusters, Intern. J. Mod. Phys. B 21, p. 2439 (2007).

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POPULATION TRANSFER PROCESSES: FROM ATOMS TO CLUSTERS AND BOSE-EINSTEIN CONDENSATE V. O. NESTERENKO∗,1 , F. F. DE SOUZA CRUZ2 , E. L. LAPOLLI2 , and P.-G. REINHARD3 1

Labor. Theor. Phys., Joint Inst. for Nuclear Research, Dubna, Moscow region, 141980, Russia, ∗ e-mail:[email protected] 2

3

Department of Physics, Federal University Santa Catarina, Florianopolis, SC, 88040-900, Brasil

Inst. of Theoretical Physics II, University of Erlangen-N¨ urnberg, D-91058, Erlangen, Germany We demonstrate that some populations transfer methods proposed for atoms and simple molecules can be successfully implemented for exploration of electronic spectra in light atomic clusters and transport of Bose–Einstein condensate. Keywords: Population transfer; STIRAP; atomic clusters; Bose–Einstein condensate.

1. Introduction The modern quantum optics delivers various methods for the population transfer (PT) between the states which cannot be related by a direct dipole transition.1 Two-photon methods, where non-dipole transfer is provided by two dipole transitions via an intermediate state, are especially spectacular. These methods range from different versions of Raman scattering to fascinating adiabatic schemes, like Stimulated Raman Adiabatic Passage (STIRAP),1,2 which promise a complete PT. So far these methods were mainly applied to atoms and simple molecules.1,2 An intriguing question is if they may be implemented to systems like atomic clusters and Bose–Einstein condensate (BEC)? The answer is not trivial. In clusters the PT, especially adiabatic ones, can be hampered by extremely short (10 − 102 fs) life-times of electronic levels, undesirable competition with plasmon modes and strong dynamical Stark shifts caused by intense lasers.3,4 As for BEC, the transfer between its components or BEC transport in a multi-well trap can be gravely affected by the atomic interaction.5 BEC transport driven by two-photon methods can be justified by a striking similarity between two-photon and tunneling schemes5 (compare plots a) and b) in Fig. 1). Both them describe the coupling between three items (states or wells) and fulfill the similar equations. Time-dependent pump and Stokes Rabi frequencies, ΩP (t), ΩS (t), describing couplings between initial/intermediate and intermediate/target states are obviously counterparts of penetration matrix elements for

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barriers separating wells in the trap. Temporal dependence of the penetrations can be simulated by the time-dependent tuning separations between the walls. In this contribution we will present examples of implementation of some PT methods to atomic clusters and BEC. In the latter case, the STIRAP transport of the BEC in the circular system of the wells (Fig. 1c) will be considered.

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2. Results and Discussion In Fig. 2 the off-resonant stimulated Raman scattering (ORSR) is applied for population of the infrared quadrupole electron-hole state at 0.75 eV in axially-deformed 4 atomic cluster Na+ 11 . The transfer is done via an intermediate dipole electron-hole state at 1.35 eV. The couplings are provided by the pump and Stokes pulses with the frequencies detuned off the resonance with the intermediate state so as to prevent its population and hence to minimize the losses. The calculations were performed within the Kohn–Sham time-dependent Hartree–Fock method,6 see details in Ref. 4. As is seen from the figure, that we get persistent quadrupole oscillation (right-top

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plot) while the dipole signal exists only during the coinciding pump and Stokes pulses (right-bottom plot). The left plots exhibit the responses in the frequency domain. It is seen that just the target E2 state at 0.75 eV is mainly excited while other E2 and E1 excitations are negligible. The population can be detected via the photoionization as described in.3 Exploration of infrared electron-hole states in deformed light clusters is important since these states deliver a direct access to the mean field spectra of valence electrons, both below and above the Fermi level. Let us now consider application of STIRAP for BEC transport. STIRAP is one of the most effective and fascinating PT methods.1,2 It assumes the counterintuitive sequence of overlapped pulses, when the Stokes pulse precedes the pump one. The adiabatic transfer takes place during the overlap time. The method deals with a dark adiabatic state which coincides with the initial state |1 > at the beginning and with the target state |3 > at the end. The intermediate state |2 > is not involved to the dark state at all, which allow to avoid losses and provide the complete (100%) population transfer. We consider STIRAP population transfer of BEC between 3 wells forming a cyclic configuration with the couplings Ω12 (t), Ω23 (t), and Ω31 (t), see Fig. 1c). The system of Gross–Pitaevskii equations for the order parameters ψ k (t) = p Nk (t)N e−iφk (t) , k = 1, 2, 3, is cast to the form N˙ k = −

3 X j

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for unknown time-dependent populations Nk (t) and phases φk (t). Following,7 we ¯k = Ek /2K, Λkj = U N and the scaled time 2Kt → t, where Ek use the notation E 2K are ground states energies of the wells, N is the total number of atoms, U is the interaction between the atoms inside the wells, and K is the coupling amplitude. Then the canonical transformation to the population imbalances z1 = N2 −N1 , z2 = N3 − N2 , z3 = N and phase differences θ1 = 1/3(−2φ1 + φ2 + φ3 ), θ2 = 1/3(−φ2 − φ2 + 2φ3 ), θ3 = 1/3(φ1 + φ2 + φ3 ) is done to remove integral of motion z3 and to reduce the problem to a system of 4 differential equations.8 Fig. 3 demonstrates a robust STIRAP transfer at Λ = 0, i.e. without the interaction and related non-linearity. Being initially (t=0) in the well 1 (N1 = 1, N2 = N3 = 0), BEC undergoes the sequence of full population transfers, 1 → 3, 3 → 2 and 2 → 1 and finally fully returns to the initial well 1 but already with another phase. The right panels show that STIRAP survives at modest interaction and detuning (Λ, ∆ < 0.4) but ruins at their values (in Ref. 5 STIRAP falls down already at Λ = 0.2). At even stronger interaction Λ = 0.7, we get a new regime: direct population transfer to the neighbor well, 1 → 2. In summary, application of particular population transfer methods, ORSR for clusters and STIRAP for BEC, was demonstrated. The former allows to get the

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electronic spectra which in turn are sensitive indicators of various cluster properties. The latter can be used for the circular transport of BEC between the coupled wells and, as BEC is coherent, for investigation of geometric phases and related physics. Acknowledgments This work was supported by the CAPES (Brazil) grant PVE 0067-11/2005 DFG grant RE 322/11-1, grant of University of Paul Sabatier (Toulouse, France) and Heisenberg-Landau (Germany-BLTP JINR) grants for 2005-2007 years. References 1. N.V. Vitanov et al., Adv. Atom. Mol. Opt. Phys., 46, 55 (2001). 2. K. Bergmann, H. Theuer, and B.W. Shore, Rev. Mod. Phys. 70, 1003 (1998). 3. V.O. Nesterenko, P.-G. Reinhard, Th. Halfmann, and L.I. Pavlov, Phys. Rev. A, 73 021201 (2006). 4. V.O. Nesterenko, P.-G. Reinhard, Th. Halfmann, and E. Suraud, J. Phys. B 39, 3905 (2006). 5. E.M. Graefe, H.J.Korsch, and D. Witthaut, Phys. Rev. A 73, 013617 (2006). 6. P.-G. Reinhard and E. Suraud, Introduction to Cluster Dynamics, (Wiley-VCH, Berlin, 2003).

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7. A. Smerzi et al, Phys. Rev. Lett. 79, 4950 (1997). 8. E.L. Lapolli, V.O. Nesterenko and F.F. de Souza Cruz, to be published.

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GENERALIZED ENTANGLEMENT IN STATIC AND DYNAMIC QUANTUM PHASE TRANSITIONS Shusa DENG and Lorenza VIOLA∗ Department of Physics and Astronomy, Dartmouth College, Hanover, NH 03755, USA ∗ E-mail: [email protected] Gerardo ORTIZ Department of Physics, Indiana University, Bloomington, IN 47405, USA E-mail: [email protected] We investigate a class of one-dimensional, exactly solvable anisotropic XY spin-1/2 models in an alternating transverse magnetic field from an entanglement perspective. We find that a physically motivated Lie-algebraic generalized entanglement measure faithfully portraits the static phase diagram — including second- and fourth-order quantum phase transitions belonging to distinct universality classes. In the simplest time-dependent scenario of a slow quench across a quantum critical point, we identify parameter regimes where entanglement exhibits universal dynamical scaling relative to the static limit. Keywords: Entanglement; quantum phase transitions; quantum information science.

1. Introduction Developing methodologies for probing, understanding, and controlling quantum phases of matter under a broad range of equilibrium and non-equilibrium conditions is a central goal of condensed-matter physics and quantum statistical mechanics. Since novel forms of matter tend to emerge in the deep quantum regime where thermal effects are frozen out, a key prerequisite is to obtain an accurate theoretical understanding of zero-temperature quantum phase transitions (QPTs).1 Aside from its broad conceptual significance, such a need is heightened by the growing body of experimental work which is being performed at the interface between material science, quantum device technology, and experimental implementations of quantum information processing (QIP). Following the experimental realization of the Bose– Hubbard model in a confined 87 Rb Bose–Einstein condensate and the spectacular observation of the superfluid-to-Mott-insulator QPT,2 ultracold atoms are enabling investigations into strongly interacting many-body systems with an unprecedented degree of control and flexibility — culminating in the observation of topological defects in a rapidly quenched spinor Bose–Einstein condensate.3 Remarkably, the

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occurrence of a QPT influences physical properties well into the finite-temperature regime where real-world systems live, as vividly demonstrated by the measured low-temperature resistivity behavior in heavy-fermion compounds.4 From a theoretical standpoint, achieving as a complete and rigorous quantummechanical formulation as desired is hindered by the complexity of quantum correlations in many-body states and dynamical evolutions. Motivated by the fact that QIP science provides, first and foremost, an organizing framework for addressing and quantifying different aspects of “complexity” in quantum systems, it is natural to ask: Can QIP concepts and tools contribute to advance our understanding of many-body quantum systems? In recent years, entanglement theory has emerged as a powerful bridging testbed for tackling this broad question from an informationphysics perspective. On one hand, entanglement is intimately tied to the inherent complexity of QIP, by constituting, in particular, a necessary resource for computational speed-up in pure-state quantum algorithms.5 On the other hand, critically reassessing traditional many-body settings in the light of entanglement theory has already resulted in a number of conceptual, computational, and information-theoretic developments. Notable advances include efficient representations of quantum states based on so-called projected entangled pair states,6 improved renormalization-group methods for both static 2D and time-dependent 1D lattice systems,7 as well as rigorous results on the computational complexity of such methods and the solvability properties of a class of generalized mean-field Hamiltonians.8 In this work, we focus on the problem of characterizing quantum critical models from a Generalized Entanglement (GE) perspective,9,10 by continuing our earlier exploration with a twofold objective in mind: first, to further test the usefulness of GE-based criticality indicators in characterizing static quantum phase diagrams with a higher degree of complexity than considered so far (in particular, multiple competing phases); second, to start analyzing time-dependent, non-equilibrium QPTs, for which a number of outstanding physics questions remain. In this context, special emphasis will be devoted to establish the emergence and validity of universal scaling laws for non-equilibrium observables. 2. Generalized Entanglement in a Nutshell 2.1. The need for GE Because a QPT is driven by a purely quantum change in the many-body groundstate correlations, the notion of entanglement appears naturally suited to probe quantum criticality from an information-theoretic standpoint: What is the structure and role of entanglement near and across criticality? Can appropriate entanglement measures detect and classify quantum critical points (QCPs) according to their universality properties? Extensive investigations have resulted in a number of suggestive results, see e.g. Ref. 11 for a recent review. In particular, pairwise entanglement, quantified by so-called concurrence, has been found to develop distinctive singular behavior at criticality in the thermodynamic limit, universal scaling laws

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being obeyed in both 1D and 2D systems. Additionally, it has been established that the crossing of a QCP point is typically signaled by a logarithmic divergence of the entanglement entropy of a block of nearby particles, in agreement with predictions from conformal field theory. While this growing body of results well illustrates the usefulness of an entanglement-based view of quantum criticality, a general theoretical understanding is far from being reached. With a few exceptions, the existing entanglement studies have focused on analyzing how (i) bipartite quantum correlations (among two particles or two contiguous blocks) behave near and across a QCP under the assumption that the underlying microscopic degrees of freedom correspond to (ii) distinguishable subsystems (iii) at equilibrium. GE provides an entanglement framework which is uniquely positioned to overcome the above limitations, while still ensuring consistency with the standard “subsystem-based” entanglement theory in well-characterized limits.9,10,12 Physically, GE rests on the idea that entanglement is an observer-dependent concept, whose properties are determined by the expectations values of a distinguished subspace of observables Ω, without reference to a preferred decomposition of the overall system into subsystems. The starting point is to generalize the observation that standard entangled pure states of a composite quantum system look mixed relative to an “observer” whose knowledge is restricted to local expectation values. Consider, in the simplest case, two distinguishable spin-1/2 subsystems in a singlet (Bell) state, |Belli =

| ↑iA ⊗ | ↓iB − | ↓iA ⊗ | ↑iB √ , 2

(1)

defined on a tensor-product state space H = HA ⊗ HB . First, the statement that |Belli is entangled — |Belli cannot be expressed as |ψiA ⊗ |ϕiB for arbitrary |ψiA ∈ HA , |ϕiB ∈ HB — is unambiguously defined only after a preferred tensorproduct decomposition of H is fixed: Should the latter change, so would entanglement in general.12 Second, the statement that |Belli is entangled is equivalent to the property that (either) reduced subsystem state — as given by the partial trace operP ation, ρA = TrB {|BellihBell|} — is mixed, Tr{ρ2A } = 1/2(1 + α=x,y,z hσαA i2 ) < 1, in terms of expectations of the Pauli spin-1/2 matrices σαA acting on A. To the purposes of defining GE, the key step is to realize that a meaningful notion of a reduced state may be constructed for any pure state |ψi ∈ H without invoking a partial trace, by specifying such a reduced “Ω-state” as a list of expectations of operators in the preferred set Ω. The fact that the space of all Ω-states is convex then motivates the following:9 Definition (Pure-state GE). A pure state |ψi ∈ H is generalized unentangled relative to Ω if its reduced Ω-state is pure, generalized entangled otherwise. For applications to quantum many-body theories, two major advantages emerge with respect to the standard entanglement definition: first, GE is directly applicable to both distinguishable and indistinguishable degrees of freedom, allowing to naturally incorporate quantum-statistical constraints; second, the property of a many-body state |ψi to be entangled or not is independent on both the choice of

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“modes” (e.g. position, momentum, etc.) and the operator language used to describe the system (spins, fermions, bosons, etc.) — depending only on the observables Ω which play a distinguished physical and/or operational role. 2.2. GE by example For a large class of physical systems, the set of distinguished observables Ω may be identified with a Lie algebra consisting of Hermitian operators, Ω ' h, which generates a corresponding distinguished unitary Lie group via exponentiation, h 7→ G = eih . While the assumption of a Lie-algebraic structure is not necessary for the GE framework to be applicable,9,12 it has the advantage of both suggesting simple GE measures and allowing a complete characterization of generalized unentangled states. In particular, a geometric measure of GE is given by the square length (according to the trace norm) of the projection of |ψihψ| onto h:

Definition (Relative purity). Let {O` }, ` = 1, . . . , M, be a Hermitian, orthogonal basis for h, dim(h) = M . The purity of |ψi relative to h is given by Ph (|ψi) = K

M X `=1

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where K is a global normalization factor chosen so that 0 ≤ Ph ≤ 1.

Notice that Ph is, by construction, invariant under group transformations, that is, Ph (|ψi) = Ph (G|ψi), for all G ∈ G, as desirable on physical grounds. If, additionally, h is a semi-simple Lie algebra irreducibly represented on H, generalized unentangled states coincide9 with generalized coherent states (GCSs) of G, that is, they may be seen as “generalized displacements” of an appropriate reference state, P |GCS({η` })i = exp(i ` η` O` )|refi. Physically, GCSs correspond to unique ground states of Hamiltonians in h: States of matter such as BCS superconductors or normal Fermi liquids are typically described by GCSs. While we refer the reader to previous work9,10,12 for additional background, we illustrate here the GE notion by example, focusing on two limiting situations of relevance to the present discussion. 2.2.1. Example 1: Standard entanglement revisited The standard entanglement definition builds on the assumption of distinguishable quantum degrees of freedom, the prototypical QIP setting corresponding to N local parties separated in real space, and H = H1 ⊗ . . . ⊗ HN . Available means for manipulating and observing the system are then naturally restricted to arbitrary local transformations, which translates into identifying the Lie algebra of arbitrary local (traceless) observables, hloc = su(dim(H1 )) ⊕ . . . ⊕ su(dim(HN )), as the distinguished algebra in the GE approach. If, for example, each of the factors H` supports a spin-1/2, hloc = span{σα` ; α = x, y, z, ` = 1, . . . , N }, and Eq. (2) yields 1 X 1 1 X hψ|σα` |ψi2 = , (3) Trρ2` − Ph loc (|ψi) = N N 2 `,α

`

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which is nothing but the average (normalized) subsystem purity. Thus, Ph loc quantifies multipartite subsystem entanglement in terms of the average bipartite entanglement between each spin and the rest. Maximum local purity, Ph = 1, is attained if and only if the underlying state is a pure product state, that is, a GCS of the local unitary group Gloc = SU (2)1 ⊗ . . . ⊗ SU (2)N . 2.2.2. Example 2: Fermionic GE Consider a system of indistinguishable spinless fermions able to occupy N modes, which could for instance correspond to distinct lattice sites or momentum modes, and are described by canonical fermionic operators cj , c†j on the 2N -dimensional Fock space HF ock . Although the standard definition of entanglement can be adapted to the distinguishable-subsystem structure associated with a given choice of modes (resulting in so-called “mode entanglement”), privileging a specific mode description need not be physically justified, especially in the presence of many-body interactions.13 These difficulties are avoided in the GE approach by associating “generalized local” resources with number-preserving bilinear fermionic operators, which identifies the unitary Lie algebra u(N ) = span{c†j cj ; 1 ≤ i, j ≤ N } as the distinguished observable algebra for fermionic GE. Upon re-expressing u(N ) in terms of an orthogonal Hermitian basis of generators, Eq. (2) yields Pu(N ) (|ψi) =

N N i 4 X † 2 X h † hcj ck + c†k cj i2 − hc†j ck − c†k cj i2 + hc cj − 1/2i2 . (4) N N j=1 j j 0, γ = 1 corresponds to the Ising model in a alternating transverse field recently analyzed in Ref. 15. While full detail will be presented elsewhere,17 an exact solution for the energy spectrum of the above Hamiltonian may be obtained by generalizing the basic steps used in the standard Ising case,14 so that to account for the existence of a twosite primitive cell introduced by the alternation. By first separately applying the Jordan–Wigner mapping to even and odd lattice sites,16 and then using a Fourier transformation to momentum space, Hamiltonian (6) may be rewritten as: n o X † X ˆ k Aˆk , K+ = π , 3π , . . . , π − π Aˆk H , Hk = H= N N 2 N k∈K+

k∈K+

ˆ k is a four-dimensional Hermitian matrix, and Aˆ† = (a† , a−k , b† , b−k ) is a where H k k k vector operator, a†k (b†k ) denoting canonical fermionic operators that create a spinless fermion with momentum k for even (odd) sites, respectively. Thus, the problem ˆ k , for k ∈ K+ . If k,1 , k,2 , k,3 , k,4 , with reduces to diagonalizing each of matrices H ˆ k , then k,1 ≤ k,2 ≤ 0 ≤ k,3 ≤ k,4 denote the energy eigenvalues of H X † Hk = k,n γk,n γk,n , n=1,...,4

† γk,n , γk,n

where are quasi-particle excitation operators for mode k in the nth band. At T = 0, the k,1 and k,2 bands are occupied, whereas k,3 and k,4 are empty, P thus the ground-state energy EGS = k∈K+ (k,1 + k,2 ), with k,1 < 0, k,2 ≤ 0. By denoting with |vaci the fermionic vacuum, and by exploiting the symmetry properties of the Hamiltonian, the many-body ground state may be expressed in Q the form |ΨiGS = k∈K + |Ψk i, with (1) (2) (3) (4) (5) (6) |Ψk i = uk +uk a†k a†−k +uk b†k b†−k +uk a†k b†−k +uk a†−k b†k +uk a†k a†−k b†k b†−k |vaci,

(7) P6 (a) 2 for complex coefficients determined by diagonalizing Hk , with |u | = 1. a=1 k Since QPTs are caused by non-analytical behavior of EGS , QCPs correspond to zeros (hc , δc , γc ) of k,2 . The quantum phase boundaries are determined by the following pair of equations: h2 = δ 2 + 1; δ 2 = h2 + γ 2 . The resulting anisotropic quantum phase diagram is showed in Fig. 1 where, without loss of generality, we set γ = 0.5. Quantum phases corresponding to disordered (paramagnetic, PM) behavior, dimer order (DM), and ferromagnetic long-range order (FM) emerge as depicted. In the general case, the boundaries between FM and PM phases, as well as between FM and DM phases, are characterized by second-order broken-symmetry QPTs. Interestingly, however, EGS develops weak singularities at (hc , δc , γc ) = (0, δ = ±γ) ,

(hc , δc , γc ) = (±1, δ = 0) ,

(8)

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Phase diagram of the spin-1/2 XY alternating Hamiltonian given in Eq. (6) with γ = 0.5.

where fourth-order broken-symmetry QPTs occur along the paths approaching the QCPs (Fig. 1, dashed-dotted lines). In the isotropic limit (γ = 0), an insulator-metal Lifshitz QPT occurs, with no broken-symmetry order parameter. For simplicity, we shall restrict to broken-symmetry QPTs in what follows, thus γ > 0. Standard finite-size scaling analysis reveals that new quantum critical behavior emerges in connection with the alternating fourth-order QCPs in Eq. (8).15 Thus, in addition to the usual Ising universality class, characterized by critical exponents ν = 1, z = 1, an alternating universality class occurs, with critical exponents ν = 2, z = 1. The key step toward applying GE as a QPT indicator is to identify a (Lie) algebra of observables whose expectations reflect the changes in the GS as a function of the control parameters. It is immediate to realize that Hamiltonian Eq. (6), once written in fermionic language, is an element of the Lie algebra so(2N ), which includes arbitrary bilinear fermionic operators. As a result, the GS is always a GCS of so(2N ), and GE relative to so(2N ) carries no information about QCPs. However, the GS becomes a GCS of the number-conserving sub-algebra u(N ) in both the fully PM and DM limit. This motivates the choice of the fermionic u(N )-algebra discussed in Example 2 as a natural candidate for this class of systems. Taking advantage of the symmetries of this Hamiltonian, the fermionic purity given in Eq. (4) becomes: i 8 X h † (9) |hak bk i|2 + |ha†−k b−k i|2 Pu(N ) = N k∈K+ i 4h † + hak ak − 1/2i2 + ha†−k a−k − 1/2i2 + hb†k bk − 1/2i2 + hb†−k b−k − 1/2i2 N Analytical results for Pu(N ) are only available for δ = 0, where GE sharply detects the PM-FM QPT in the XY model.10 Remarkably, ground-state fermionic GE still faithfully portraits the full quantum phase diagram with alternation. First, derivatives of Pu(N ) develop singular behavior only at QCPs, see Fig. 2 (left). Furthermore, GE exhibits the correct scaling properties near QCPs.10 By taking a Taylor expan-

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sion, Pu(N ) (h) − Pu(N ) (hc ) ∼ ξ −1 ∼ (h − hc )ν , where ξ is the correlation length, the static critical exponent ν may be extracted from a log-log plot of Pu(N ) for both the Ising and the alternating universality class, as demonstrated in Fig. 2 (right).

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Fig. 2. Pu(N ) as a static QPT indicator. Left panel: Purity and rescaled purity derivative vs magnetic field strength. Inset: second derivative for N = 1000, 2000, 4000, 8000 (top to bottom). Right panel: Determination of ν for both the alternating and Ising (inset) universality class.

3.2. Dynamic QPTs While the above studies provide a satisfactory understanding of equilibrium quantum critical properties, dynamical aspects of QPTs present a wealth of additional challenges. To what extent can non-equilibrium properties be predicted by using equilibrium critical exponents? The simplest dynamical scenario one may envision arises when a single control parameter is slowly changed in time with constant speed τq > 0, that is, g(t) − gc = (t − tc )/τq , so that a QCP is crossed at t = tc (tc = 0 without loss of generality). The typical time scale characterizing the response of the system is the relaxation time τ = ~/∆ ∼ |g(t) − gc |−zν , ∆ being the gap between the ground state and first accessible excited state and z the dynamic critical exponent.1 Since the gap closes at QCPs in the thermodynamic limit, τ diverges even for an arbitrarily slow quench, resulting in a critical slowing-down. According to the socalled Kibble-Zurek mechanism (KZM),18 a crossover between an (approximately) adiabatic regime to an (approximately) impulse regime occurs at a freeze-out time −tˆ, whereby the system’s instantaneous relaxation time matches the transition rate, τ (tˆ) = |(g(tˆ) − gc )/g 0 (tˆ)| ,

tˆ ∼ τqνz/(νz+1) ,

resulting in a predicted scaling of the final density of excitations as n(tF ) ∼ τq−ν/(νz+1) .

(10)

While agreement with the above prediction has been verified for different quantum systems,19 several key points remain to be addressed: What are the required physical

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ingredients for the KZM to hold? What features of the initial (final) quantum phase are relevant? How does dynamical scaling reflect into entanglement? In our model, the time-evolved many-body state at instant time t, |Φ(t)i = Q k∈K + |Φk (t)i, may still be expressed in the form of Eq. (7) for time-dependent (a) coefficients uk (t), a = 1, . . . , 6, computed from the solution of the Schr¨ odinger equation. The final excitation density is then obtained from the expectation value of the appropriate quasi-particle number operator over the final time, n(tF ) =

X † 1 † hΦ(tF )| (γk,3 γk,3 + γk,4 γk,4 ) |Φ(tF )i . N k∈K+

As shown in Fig. 3 (left), the resulting value agrees with Eq. (10) over an appropriate τq -range irrespective of the details of the QCP and the initial (final) quantum phase: n(tF )Ising ∼ τq−1/2 ,

n(tF )Alternating ∼ τq−2/3 .

More remarkably, however, our results indicate that scaling behavior holds throughout the entire time evolution (see Fig. 3, right), implying the possibility to express the time-dependent excitation density as: t − t c , n(t) = τq−ν/(νz+1) F tˆ

where F is a universal scaling function. Numerical results support the conjecture that similar universal dynamical scaling may hold for arbitrary observables. 17 In particular, fermionic GE obeys scaling behavior across the entire dynamics provided that the amount relative to the instantaneous ground state |Ψ(t)iGS is considered: t − t c ∆Pu(N ) (t) ≡ Pu(N ) (|Φ(t)i) − Pu(N ) (|Ψ(t)iGS ) = τq−ν/(νz+1) G , ˆ t for an appropriate scaling function G, see Fig. 4.

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(t−tc)/τ2/3 Q

Fig. 3. Dynamical scaling of the excitation density. Left panel: log-log plot for Ising universality class. Right panel: alternating universality class, with log-log scaling plot in the inset.

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Dynamical scaling of Pu(N ) for the alternating and the Ising (inset) universality class.

4. Conclusion In addition to further demonstrating the usefulness of the GE notion toward characterizing static quantum critical phenomena, we have tackled the study of timedependent QPTs in a simple yet illustrative scenario. Our analysis points to the emergence of suggestive physical behavior and a number of questions which deserve to be further explored. In particular, while for gapped systems as considered here, the origin of the observed universal dynamical scaling is likely to be rooted in the existence of a well-defined adiabatic (though non-analytic) limit — as independently investigated in Ref. 20, a rigorous understanding remains to be developed. We expect that a GE-based perspective will continue to prove valuable to gain additional insight in quantum-critical physics. Acknowledgments It is a pleasure to thank Rolando Somma and Anatoli Polkovnikov for useful discussions and input. S. D. gratefully ackowledges partial support from Constance and Walter Burke through their Special Project Fund in Quantum Information Science. References 1. S. Sachdev, Quantum Phase Transitions (Cambridge UP, Cambridge, 1999). 2. M. Greiner, O. Mandel, T. Esslinger, T. W. H¨ ansch, and I. Bloch, Nature 415, p. 39 (2002). 3. L. E. Sadler, J. M. Higbie, S. R. Leslie, M. Vengalattore, and D. M. Stamper-Kurn, Nature 443, p. 312 (2006). 4. P. Gegenwart et al., Phys. Rev. Lett., 89, p. 056402 (2002). 5. R. Jozsa and N. Linded, Proc. Roy. Soc. London A 459, p. 2001 (2003); G. Vidal, Phys. Rev. Lett. 91, p. 147902 (2003). 6. F. Verstraete and J. I. Cirac, arXiv: cond-mat/0407066 (2004); Phys. Rev. A 70, p. 060302(R) (2004).

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7. D. Porras, F. Verstraete, and J. I. Cirac, Phys. Rev. B 73, p. 014410 (2006); G. Vidal, Phys. Rev. Lett. 93, p. 040502 (2004). 8. J. Eisert, Phys. Rev. Lett. 97, p. 260501 (2006); R. Somma, H. Barnum, G. Ortiz, and E. Knill, Phys. Rev. Lett. 97, p. 190501 (2006). 9. H. Barnum, E. Knill, G. Ortiz, and L. Viola, Phys. Rev. A, 68, p. 032308 (2003); H. Barnum, E. Knill, G. Ortiz, R. Somma, and L. Viola, Phys. Rev. Lett., 92, p. 107902 (2004). 10. R. Somma, G. Ortiz, H. Barnum, E. Knill, L. Viola, Phys. Rev. A 70, p. 042311 (2004); R. Somma, H. Barnum, E. Knill, G. Ortiz, and L. Viola, Int. J. Mod. Phys. B 20, 2760 (2006). 11. L. Amico, R. Fazio, A. Osterloh, and V. Vedral, arXiv: quant-ph/0703044 (2007). 12. L. Viola and H. Barnum, arXiv:quant-ph/0701124 (2007), and references therein. 13. M. Kindermann, Phys. Rev. Lett. 96, p. 240403 (2006). 14. P. Pfeuty, Ann. Phys. 57, p. 79 (1970); E. Barouch, B. M. McCoy, and M. Dresden, Phys. Rev. A 2, p. 1075 (1970). 15. O. Derzhko and T. Krokhmalskii, Czech. J. Phys. 55, p. 605 (2005); O. Derzhko, J. Richter, and T. Krokhmalskii, Phys. Rev. E 69, p. 066112 (2004). 16. K. Okamoto and K. Yasumura, J. Phys. Soc. Japan 59, p. 993 (1990). 17. S. Deng, G. Ortiz, and L. Viola, in preparation. 18. W. H. Zurek, U. Dorner, and P. Zoller, Phys. Rev. Lett. 95, p. 105701 (2005). 19. J. Dziarmaga, Phys. Rev. Lett. 95, p. 245701 (2005); F. M. Cucchietti, B. Damski, J. Dziarmaga, and W. H. Zurek, Phys. Rev. A 75, p. 023603 (2007). 20. A. Polkovnikov, Phys. Rev. B, 72, p. 161201 (2005); A. Polkovnikov and V. Gritsev, arXiv:cond-mat/0706.0212 (2007).

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ENTANGLEMENT PERCOLATION IN QUANTUM NETWORKS: HOW TO ESTABLISH LARGE DISTANCE QUANTUM CORRELATIONS? A. ACÍN∗ and M. LEWENSTEIN ICREA and ICFO–Institut de Ci` encies Fot` oniques Mediterranean Technology Park, E-08860 Castelldefels, Barcelona, Spain ∗ E-mail: [email protected] www.icfo.es J. I. CIRAC Max-Planck–Institut f¨ ur Quantenoptik Hans-Kopfermann-Str. 1, D-85748 Garching, Germany E-mail: [email protected] Quantum communication networks consist of N distant nodes sharing a quantum state. By means of local operation in each node assisted by classical communication, the nodes try to transform the initial state into perfect quantum correlations, that later will be used to perform a quantum information task, such as quantum teleportation or quantum cryptography. Given a network, defined by a geometry of nodes and connections, it is crucial to understand whether it is possible to establish long-distance quantum correlations, in the sense that the correlations between two end points of the network do not decrease exponentially with the number of intermediate connections. In this contribution, we present our recent findings on the distribution of entanglement through quantum networks. In the case of one-dimensional chains of connected quantum systems, the results are hardly surprising: a non-exponential decay is possible only when the entanglement in the connections between nodes is larger than a maximally entangled state of two qubits. The picture becomes much richer and interesting for networks of dimension larger than one: long-distance correlations can be established even when the connecting nodes are not maximally entangled. Actually, the problem of establishing maximally entangled states between nodes is related to classical percolation in statistical mechanics. We show, then, that statistical concepts, such as percolation and phase transitions, can be used to optimize the entanglement distribution through quantum networks. Remarkably, the quantum features allow going beyond the known results for classical percolation, giving rise to a new type of critical phenomenon that we call entanglement percolation. Keywords: Entanglement; quantum communication; phase transitions; percolation theory.

1. Introduction In the recent years quantum information science (QIS) has made an important impact on condensed matter physics, and, more generally, statistical physics. Paradigm examples of this impact concern, for instance, the role of entanglement (i.e. genuine

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quantum correlations) in quantum phase transitions,1–3 or development of novel, quantum information based codes for the efficient numerical simulation of manybody systems.4,5 Statistical physics is making also an emerging inverse impact on QIS. Examples include, among others, the possibility of phase transitions in Isinglike linear networks,6,7 or proposals for employing topological order in many-body systems for robust quantum computation.8–10 In this contribution, we discuss how to apply the ideas of statistical physics in a quantum communication scenario. Quite generally, it is expected that quantum communication will be realized in Quantum Networks (QN) consisting of distant nodes, where the correlations among the nodes are described by a global quantum state. A standard problem in this scenario is to study how the initial state of the network can be processed by Local Operations at the nodes, and Classical Communication between them (LOCC). Perhaps, one of the most important tasks in such situations will consist in establishing genuine quantum correlations at large distances within the network. Several authors have addressed this challenge previously and came up with various solutions, such as, for instance, quantum repeaters11 that employ quantum distillations protocols. While the entanglement in reduced subsystems remains typically short range and local even at criticality,1,2 one can achieve large distance entanglement by LOCC.12 In general, the problem of establishing large distance entanglement in QNs is very complicated and involve very many parameters describing network architecture, connections, etc. Here, we restrict our considerations to QNs with nearest neighbor connections only, presented schematically in Fig. 1, in 1D and 2D. Our QNs consist of a (typically regular) lattice of nodes connected by entangled, but not necessarily maximally entangled pure-state pairs. The goal is to find the LOCC strategy to maximize the entanglement, according to a fixed figure of merit, between two “extremal” nodes, or ports of the QN. In this contribution, we review our main findings on the problem of establishing entanglement through QNs of different dimension.13 We show that the distribution of entanglement in this scenario displays some counter-intuitive effects. For 1D chains we show the exponential decay of probability of establishing a maximally entangled state between the ends of the chain as the chain length grows. However, a certain figure of merit, i.e. singlet conversion probability remains the same for a network consisting of a single pair, and two pairs (Fig. 1a). We study also LOCC strategies in small networks in 2D. In 2D and higher dimensions we connect this problem to classical percolation. We show that using the optimal singlet conversion strategy with probabilities larger that critical probability of percolation, it is possible to establish with certainty a perfect quantum channel for large networks (in the thermodynamic limit). Similarly, a perfect quantum channel between specified nodes can be established with finite (distance independent) probability. We call this effect classical entanglement percolation (CEP). Amazingly, using other strategies, it is possible to go beyond CEP, and achieve a perfect quantum channel in the situations when CEP is not possible. We call this effect entanglement percolation.

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The problem formulated in this paper, although originating in QIS, has many direct links to statistical physics and opens very many novel and stimulating challenges for interdisciplinary research. 2. Establishing Entanglement in 1D Chains We first study how entanglement can be established in 1D chains. As shown in Fig. 1, two nodes, A and B, are connected by N repeaters. Each of the bonds corresponds d d to an entangled pure state |ϕk i ∈ C ⊗ C . The goal is to maximize the averaged entanglement between the two end nodes by measurements in the N repeaters assisted by local communication. We denote any of these measurement strategies by M. As discussed above, this maximization depends on the entanglement parameter, that is, the figure of merit, to be maximized. Denote by f this quantity. The goal is to determine the maximum of this quantity over all LOCC measurement strategies on the N repeaters of the 1D chain, X (1) pµ f (ψµAB ), f1,N = sup M

µ

where the index µ denote the results of the LOCC measurement, pµ is the corre sponding probability, and ψµAB denotes the resulting state between Alice and Bob when measurement outcome µ is obtained. We consider the case in which the connecting states are of two qubits, i.e. |ϕi ∈ 2 2 C ⊗ C . Given a generic two-qubit state, there always exists a choice of local bases by Alice and Bob such that p p (2) |ϕi = λ1 |00i + λ2 |11i , where λ1 + λ2 = 1, while |0i and |1i define a basis, not necessarily the same, in each local space. This is known as the Schmidt decomposition and λ1 and λ2 are the Schmidt coefficients. We can impose λ1 ≥ λ2 without loss of generality. All the entanglement properties of the two-qubit state |ϕi are given by its Schmidt coefficients.

2.1. Concurrence As a first figure of merit we take the concurrence. It is a measure of entanglement that, in the case of pure states of two qubits, is proportional to the modulus of the determinant of the reduced state ρA = TrB |ϕihϕ|.14 It is relatively simple to see that the maximization of the averaged concurrence is given by X (3) C1,N = sup 2| det(ϕ1 Mµ1 ϕ2 . . . MµN ϕN +1 )|. M

µ

Here ϕk represent 2 × 2 diagonal matrices given by the Schmidt coefficients of the states |ϕk i. Mµk are also 2 × 2 matrices, corresponding to the pure state, |µk i, associated to the measurement result µk of the k-th repeater, that is |µk i =

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(a) A

(b)

R

B

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R2

B

RN

...

... ...

...

...

...

...

...

...

(c)

...

...

...

...

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... Fig. 1. Quantum networks in one and two dimensions. (a) The simplest one-repeater configuration, that is generalized in (b). (c) Square lattice in 2D. All the bonds correspond to a pure state of two qubits, |ϕi.

P

i,j (Mµk )ij

|iji. Note that the computational basis i and j in the previous expressions are the Schmidt basis for the states |ϕk i and |ϕk+1 i entering the repeater k. Moreover, X |µk ihµk | = 11 ⊗ 11, (4) µk

d

d

which follows from the fact that the states |µk i define a measurement in C ⊗ C . Using the fact that det(AB) = det(A) det(B), we can rewrite the previous maximization as C1,N =

N Y

k=1

|2 det(ϕk )| sup M

X µ

|2 det(Φ+ Mµ1 Φ+ . . . MµN Φ+ )|,

(5)

√ where Φ+ = 11/ 2 is the matrix √ corresponding to the two-qubit maximally entangled state |Φ+ i = (|00i+|11i)/ 2, which is equivalent, in terms of entanglement, to a single state of two spin-one-half particles. Thus, the second term in this formula can be identified with the localizable concurrence in the situation where all states

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are maximally entangled, which is equal to one. Thus we have C1,N =

N Y

k=1

|2 det(ϕk )|.

(6)

Note that |2 det(ϕk )| if and only if |ϕk i is maximally entangled. Not surprisingly, the singlet fidelity decreases exponentially with the number of repeaters unless the connecting states are maximally entangled. 2.2. Other figures of merit All the previous discussion was based on the singlet fidelity or, equivalently, the two-qubit concurrence. However, one can consider other figures of merit, that may be more meaningful for some specific scenarios, as it becomes clearer below. In this section, we simply consider two different figures of merit and study their optimization for the simplest one-repeater configuration consisting of two identical two-qubit pure states, |ϕi. The obtained results will later be used in the 2D case. First, we consider the probability of LOCC conversion into a perfect singlet. We define the singlet conversion probability (SCP), which is the optimal probability that a maximally entangled state can be established between two given nodes. It is a known result from majorization theory that a two-qubit state (2) can be converted by LOCC into a singlet with maximal probability p ok = 2λ2 , where λ2 is the smallest Schmidt coefficient.a Consider the one-repeater configuration, see Fig. 1a, where the two states are equal and given by (2). The goal is now to apply a measurement in the repeater such that the average probability of conversion into a singlet for the resulting state between A and B, p ok 1,1 , is maximized. It is clear ok that p 1,1 cannot be larger than p ok . If this was the case, it would contradict the majorization result, by putting nodes A and R1 together. In the next lines we show that this bound can indeed be achieved. The optimal protocol turns out to be rather simple. First, entanglement swapping in the computational bases is performed. This means that a measurement in the basis ± Φ = √1 (|0i |0i ± |1i |1i) 2 ± 1 Ψ = √ (|0i |1i ± |1i |0i), (7) 2 is performed. One can see that the resulting states and probabilities are p

1

(λ1 |00i ± λ2 |11i) 2

λ21 + λ2

1 √ (|01i ± |10i) 2

p=

λ21 + λ22 2

p = λ 1 λ2 .

(8)

a Several results from majorization theory applied to entanglement are used in this work. For details, see the work by Nielsen and Vidal.15

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The average singlet conversion probability (SCP) between A and B then reads 2 2 p ok 1,1 = 2λ1 λ2 + (λ1 + λ2 ) × 2

λ21

λ2 = 2λ2 = p ok , + λ22

(9)

as announced. It is surprising that the intermediate repeater step does not imply ok a loss of SCP. Of course, p ok when N > 1. Actually, it follows from the 1,N < p ok previous reasoning for the concurrence that p 1,N should also decrease exponentially with N . Finally, we consider the optimal protocol according to a second figure of merit that we call worst-case entanglement. The idea is to find a measurement optimizing the entanglement for all the outcomes. The amount of entanglement is measured by a chosen entanglement measure, E. Then, we look for the measurement strategy such that E1,N = sup min E(ϕ(µ)), M

µ

(10)

i.e. that maximizes the minimal entanglement between A and B over all measurement outcomes. Here we only consider the case of one-repeater configuration and qubits, and we take as entanglement measure the smallest Schmidt coefficient, which is related to the SCP. The optimal measurement consists of entanglement swapping in the zx basis.16 This means that the two particles in the repeater are measured in the basis: ± Φzx = √1 (|0i |+i ± |1i |−i) 2 ± 1 Ψzx = √ (|0i |−i ± |1i |+i), (11) 2 √ where |±i = (|0i±|1i)/ 2. After some patient algebra, one can see that the resulting states for all the measurement outcomes have the same Schmidt coefficients. If the Schmidt coefficients of the initial states are λ1 ≥ λ2 , for the first state, and ν≥ ν2 , for the second, the new Schmidt coefficients read p ˜ i = 1 1 ± 1 − 16λ1 λ2 ν1 ν2 . λ (12) 2 This protocol will be used in the next section. 3. 2D Lattices The previous results for the 1D chain are quite intuitive: there is an exponential decay, with the number of repeaters, of the entanglement between the two end nodes whenever the connecting states are less entangled than the singlet. The situation becomes much more interesting in the case of 2D geometries. Note that this is the situation that naturally appears when considering quantum information networks. The scope of this section is to explore this 2D scenario. We consider lattices with different geometries where all the nodes are connected by an entangled pure state |ϕi of Schmidt coefficients λ1 ≥ λ2 , see also Fig. 1c.

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Measurement strategies for the simplest 2 × 2 lattice.

3.1. Finite lattices We start by finite 2D lattices, in particular by the simplest 2 × 2 square lattice. The goal is to determine the minimal amount of entanglement of the state |ϕi such that two of the end points of the lattice can establish a perfect channel with probability one. First, consider the sites 1 and 4, see Fig. 2.a below. Nodes 2 and 3 perform a measurement maximizing the worst-case entanglement, as explained above. Now, nodes 1 and 4 are connected by two two-qubit pure states, |ϕ(i)i and |ϕ(j)i that depend on the results i and j of the measurements by node 2 and 3, respectively. Since the aim is to establish a singlet with probability one, |ϕ(i)i ⊗ |ϕ(j)i should majorize the singlet for all i and j.15 At this point, it has to be clear why we first applied the worst-case measurement in nodes 2 and 3. One then sees that this protocol allows nodes 1 and 4 to establish a singlet with probability one when the initial state |ϕi satisfies15 1 ≤ λ1 / 0.6498. (13) 2 Thus, because of the richer geometry, sufficiently, though not maximally, entangled states are sufficient to have a perfect channel. Whether this bound can be improved is an interesting question that deserves further investigation. A similar strategy can be applied when considering nodes 1 and 2, see Fig. 2.b. There, entanglement swapping, according to the worst-case entanglement criterion, is performed at nodes 3 and 4. The same amount of entanglement, |ϕ(i)i is obtained for all measurement outcomes in nodes 3 and 4. The resulting state, |ϕ(i)i ⊗ |ϕi between 1 and 2 can be transformed into a singlet in a deterministic way whenever √ 1 5−1 ≤ λ1 ≤ ≈ 0.658. (14) 2 2 Again, even if the sites are connected by states that are not maximally entangled, it is possible to establish a perfect quantum channel between the ends with probability equal to one.

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3.2. Classical entanglement percolation Let us now move to the asymptotic large network regime (thermodynamic limit). The goal is to establish a perfect singlet between the two sides of the lattice. Using majorization strategies, it is possible to link this problem with standard percolation theory.17 Consider a lattice where the nodes are connected by pure states |ϕi that do not majorize the singlet. It is however still possible to transform this state into a singlet, the optimal probability p ok being specified by majorization theory. Using this strategy, the problem is equivalent to a bond percolation situation. In bond percolation, one has a network of nodes and distributes connections among the nodes in a probabilistic way: with probability p, an edge connecting the nodes is established, otherwise the nodes are kept unconnected. For each lattice geometry there exists a percolation threshold probability, pth , such that an infinite length path can be established through the network if and only if p > pth (see also Table 1). It is clear that by applying the majorization measurement strategy to the network consisting of pure entangled states, we map the problem into a percolation problem: with probability p ok a perfect quantum channel is established, otherwise no entanglement is left. Therefore, the threshold probabilities define the minimal amount of entanglement for the initial state such that entanglement percolation is possible. We call this measurement strategy classical entanglement percolation (CEP). In the qubit case, the minimal entanglement for classical entanglement percolation is given by 2λ2 = pth . This defines a critical entanglement for each lattice geometry. It is easy to further conclude that p ok < pth , then the probability of establishing a perfect channel decreases exponentially with distance, whereas for p ok > pth , then in the large distance limit, the probability for establishment of entanglement between the definite input-output ports is non-zero,17 independently of the distance. Classical percolation, then, allows for long distance entanglement distribution in multi-connected networks. Table 1. Bond Percolation Threshold Probabilities for some 2D lattices. Lattice

Percolation Threshold Probability

Square Triangular Honeycomb

1/2 2 sin (π/18) ≈ 0.3473 1 − 2 sin (π/18) ≈ 0.6527

3.3. Quantum entanglement percolation The natural question is whether the thresholds defined by classical percolation theory are optimal or entanglement percolation represents a related but different theoretical problem where new bounds have to be obtained. This is of course equivalent

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Fig. 3. Each node is connected by two copies of the same two-qubit state, |ϕi. The nodes marked in (a) perform the measurement optimal according to the SCP. A triangular lattice is obtained where the SCP is the same as for the state |ϕi.

to determine whether the previous CEP measurement strategy is optimal in the asymptotic regime. Here, we show an example that goes beyond the classical percolation picture, proving that the CEP strategy is not optimal. The key ingredient for the construction of the example is the measurement strategy previously obtained for the 1D one-repeater configuration that maximizes the SCP. Consider a honeycomb lattice where each node is connected by two copies of the same two-qubit state |ϕi, see Fig. 3.a. If, as above, the Schmidt coefficients of the ⊗2 two-qubit state are λ1 ≥ λ2 , the SCP of |ϕi is given by pok = 2(1 − λ21 ).15 We choose this conversion probability slightly smaller than the percolation threshold for the honeycomb lattice (see Table 1), r π 1 + sin ≈ 0.82, (15) λ1 = 2 18

so the classical entanglement percolation strategy is useless. Now, some of the nodes, see Fig. 3.a, perform the optimal strategy for the SCP, mapping the honeycomb lattice into a triangular lattice, as shown in Fig. 3.b. Fig. 1.a What is important is that the SCP for the new bonds is exactly the same as for the initial state |ϕi, that is 2λ2 . This probability is indeed larger than the percolation threshold for the triangular lattice, since ! r π π 1 + sin 2λ2 = 2 1 − ≈ 0.358 > 2 sin . (16) 2 18 18

Thus, the nodes can now apply classical entanglement percolation strategy. This proof-of-principle construction, then, implies that the problem of entanglement distribution through quantum communication networks defines a new type of phase transition, where new threshold values have to be computed. We name this phenomenon entanglement percolation.

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4. Conclusions Entanglement distribution in 1D and, especially, 2D networks is a very interesting and almost unexplored problem, which is crucial for the future development of quantum communication networks. It defines a new type of phase transition that we call entanglement percolation. As our results show, unexpected results and deep connections with other mathematical techniques and physical problems, such as percolation theory, can be exploited. There are plenty of open questions, especially for the 2D and higher dimensional cases. One of the most fundamental questions is to determine the minimal amount of entanglement allowing some kind of full connectivity in the lattice. Our results prove that one can go beyond the classical percolation strategy. However, we were not able to establish any lower on the minimal amount of pure-state entanglement such that entanglement percolation is possible in a 2D lattice. Another interesting question, especially from a practical point of view, is to identify the optimal strategies for finite lattices. Progress in this direction has recently been obtained.16 But, perhaps, the most relevant open question is to extend these results to the mixed-state case, constructing an example of a network with noisy bonds where entanglement percolation is possible. Acknowledgments We thank D. Cavalcanti, S. Perseguers and J. Wehr for enlightening discussions. We acknowledge support from Deutsche Forschungsgemeinschaft (SFB 407, SPP 1116), EU IP Programme SCALA and QAP, European Science Foundation PESC QUDEDIS,and MEC (Spanish Government) under contracts FIS 2005-04627, and Consolider QOIT. References 1. A. Osterloh, L. Amico, G. Falci and R. Fazio, Nature 416, p. 608 (2002). 2. T. J. Osborne and M. A. Nielsen, Phys. Rev. A 66, p. 032110 (2002). 3. F. Verstraete, M.A. Martín-Delgado and J.I. Cirac, Phys. Rev. Lett. 92, p. 087201 (2004). 4. G. Vidal, Phys. Rev. Lett. 91, p. 147902 (2003). 5. F. Verstraete, D. Porras, and J. I. Cirac, Phys. Rev. Lett. 93, p. 227205 (2004). 6. P. T¨ orm¨ a, Phys. Rev. Lett. 81, p. 2185 (1998). 7. J. Novotny, M. Stefanak, T. Kiss, and I. Jex, J. Phys. A 38, p. 9087 (2005). 8. A.Yu. Kitaev, Annals of Physics 303, p. 2 (2003). 9. A. Micheli, G. K. Brennen and P. Zoller, Nature Physics 2, p. 341 (2006). 10. S. Das Sarma, M. Freedman and C. Nayak, Phys. Today 59, p. 32 (2006). 11. H.-J. Briegel, W. D¨ ur, J. I. Cirac and P. Zoller, Phys. Rev. Lett. 81, p. 5932 (1998). 12. F. Verstraete, M. Popp and J.I Cirac, Phys. Rev. Lett. 92, p. 027901 (2004). 13. A. Ac´ın, J. I. Cirac and M. Lewenstein, Nature Physics 3, p. 256 (2007). 14. W. K. Wootters, Phys. Rev. Lett. 80, p. 2245 (1998). 15. M. A. Nielsen and G. Vidal, Quant. Inf. Comp. 1, p. 76 (2001). 16. S. Perseguers, J. I. Cirac, J. Wehr, A. Ac´ın and M. Lewenstein, arXiv:0708.1025. 17. G. Grimmett, Percolation, (Springer, Berlin, 1999).

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PHONON-ROTON EXCITATIONS AND QUANTUM PHASE TRANSITIONS IN LIQUID 4 HE IN NANOPOROUS MEDIA Henry R. GLYDE∗ , Jonathan V. PEARCE2 , Jacques BOSSY3 , and Helmut SCHOBER4 [*] Department of Physics and Astronomy, University of Delaware, Newark, Delaware, 19716, US [2] National Physical Laboratory, Hampton Road, Teddington, TW11 0LW, U.K. [3] Institut N` eel, CNRS-UJF, BP 166, 38042 Grenoble, France [4] Institut Laue Langevin, BP 156, 38042 Grenoble, France We present measurements of the elementary phonon-roton and other excitations of liquid confined in nanoporous media using inelastic neutron scattering methods. The aim is to compare phonon-roton (P-R) and superfluid density measurements and to explore the interdependence of Bose–Einstein Condensation (BEC), P-R modes and superfluidity in helium at nanoscales and in disorder. Specifically a goal is to determine the region of temperature and pressure in which well defined phonon-roton modes exist and compare this with the superfluid phase diagram. In porous media the liquid phase is extended up to 35-40 bars. A second goal is to investigate helium at higher pressures. At low temperature and at saturated vapor pressure (SVP) (p ' 0) liquid 4 He supports well defined P-R modes in all porous media investigated to date (aerogel, xerogel, Vycor, MCM-41 and gelsil of several pore diameters). As temperature is increased at SVP, the P-R modes broaden but well defined modes exist above Tc in the normal phase, up to Tλ . The superfluid to normal transition temperature, Tc , in porous media always lies below the corresponding temperature, Tλ , in bulk helium. In liquid 4 He in 25 ˚ A and 34 ˚ A mean pore diameter gelsil under pressure and at low temperature, we observe loss of all well defined P-R modes at p = 36.3 - 36.8 bars. Yamamoto et al. have observed a possible Quantum Phase Transition (QPT) (T ≈ 0 K) from the superfluid to normal liquid at p = 34 bars in superfluid density measurements. The existence of P-R modes under pressure up to 36.3 bars and their subsequent loss supports the finding of a QPT. We discuss the implications of these results for the basic concepts of BEC and superfluidity in helium at nanoscales and in disorder. 4 He

Keywords: Bose–Einstein Condensation; elementary excitations; neutron scattering.

1. Introduction In this talk we present neutron scattering measurements of the phonon-roton (P-R) excitations of superfluid 4 He confined to nanoscales in porous media. We emphasize helium confined in gelsils under pressure. Superfluidity of confined liquid 4 He has been investigated extensively in many porous media.1–6 Recent measurements have focused on the superfluid density, ρS , in smaller pore media to enhance the effects of confinement.3–6 Our aim is to add measurements of the P-R excitations,7–17 and where possible18 of Bose–Einstein condensation (BEC), to explore the interdependence of BEC, P-R modes and superfluidity in nanoporous media. When helium is confined in porous media, the superfluid to normal transition

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temperature, Tc , is depressed below the bulk liquid value Tλ , where Tλ = 2.17 K at saturated vapor pressure (SVP) (p ' 0). The smaller the pore diameter, the lower is Tc . In Vycor Tc = 1.95 to 2.05 K1,14,19 at SVP depending on the specific Vycor sample. The mean pore diameter (mpd) of Vycor is d = 70 ˚ A with RMS deviation around the mean δd ' 6 ˚ A. In 44 ˚ A and 25 ˚ A mpd gelsil, Tc = 1.92 K and Tc ' 1.4 K at SVP, respectively.4,15 The gelsils are structurally similar to aerogels and have a broad pore size distribution with δd ' 20 ˚ A in 25 ˚ A and 34 ˚ A mpd gelsil. The temperature dependence of ρS (T ) below Tc in porous media is also different from the bulk and varies from media to media.1,2 The apparent critical exponent of ρS (T ) is predicted20 to depend on δd with Vycor having a narrow pore size distribution and an apparent exponent close to the bulk value. In addition, confinement adds disorder which is predicted21–25 to decrease the condensate fraction, n0 , and ρS . Indeed point disorder is predicted21,24 to reduce the ρS more than n0 with sufficiently strong disorder bringing ρS to zero at a finite condensate fraction.24 In addition, going back to the work of Fisher et al.,26 glass phases that have localized BEC and no superflow have been predicted. Our experiments9,10,12,14–17 certainly suggest that there is a localized BEC (glass) region between the superfluid and normal phases. The superfluid transition is associated with a crossover from extended BEC to a localized BEC glass phase.

(a)

(b) 4 He.

4 He

Fig. 1. Schematic phase diagram of (a) LHS: bulk showing superfluid and normal liquid phases with transition temperature at Tλ . (b) RHS: 4 He confined in a nanoporous media showing a suppression of the normal-superfluid transition temperature to Tc and extension of liquid phase to pressures of 3.5 - 4.0 MPa.

The liquid phase of helium is also extended to higher pressure in confinement.6,27,28 A schematic phase diagram of helium in the bulk and in a porous media is shown in Fig. 1. The liquid-solid transition is extended from pressure p = 25.3 bars in bulk helium to approximately 30–40 bars depending on the porous media.27,28 A possible Quantum Phase transition (QPT), a superfluid to normal phase transition at T ' 0 K, has been reported at p = 34 bars in a 25 ˚ A mpd gelsil.4

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As observed in a 44 ˚ A gelsil15 we find below that well defined P-R modes are not observed above p ' 36.5 bars in 25 ˚ A gelsil which supports this finding. Solidification is estimated to begin at 34.2-36.5 bars in 25 ˚ A mpd gelsil6 with liquid-solid co-existence at higher pressures. There is also broad interest in BEC, P-R modes and thermodynamic properties of bulk helium at high pressures.29–31 We begin by comparing BEC and the P-R modes observed in bulk helium and in porous media. We then turn to helium in porous media and liquid helium at higher pressure. 2. Bulk and Confined Helium Compared Historically, Bose introduced Bose statistics and Einstein32 noted that, as a consequence, a macroscopic fraction of the Bosons would condense into one single particle state at low temperature, denoted Bose–Einstein condensation (BEC). Superfluidity in bulk liquid helium was first reported in 1938 and in the same year London33 proposed that it was a manifestation of BEC. In 1947 Bogoliubov 34 showed that a weakly interacting gas of Bosons with a condensate supported phonon like excitations and was indeed a superfluid. He also showed that in the presence of BEC the single particle and density excitations (phonons) have the same phonon like energy. In modern language, BEC means macroscopic occupation of a single particle state. The wave function of this state has a magnitude and a phase, φ(r). If this state is connected and continuous across the system, then so is the phase and there is superflow across the system with velocity given by vs = (~/m)5φ. The conditions (e.g. disorder) under which superfluidity is expected to arise from BEC is discussed by Huang.22 BEC has now been spectacularly observed in dilute gases confined in traps with nearly 100 % of the atoms condensed into the lowest energy single particle state. Equally, there is a condensate in a dense Bose liquid such as liquid 4 He. However, the fraction of atoms condensed in one state, the zero momentum state in bulk superfluid helium, is small. The condensate fraction is depleted by the strong interatomic interaction in a dense fluid. The LHS of Fig. 2 shows n0 observed9 in bulk liquid 4 He at SVP. At T = 0 K, n0 (0)= 7.25 ± 0.75 % and n0 (T) goes to zero at the superfluid to normal transition temperature Tλ within precision. Path integral Monte Carlo (PIMC)36,39 and diffusion Monte Carlo (DMC)31,38,40 calculations agree well with the observed value. As pressure is applied, the density to bulk helium increases and n0 is further depleted by interactions. DMC calculations predict n0 (0)' 2% at p = 25.3 bars where bulk helium solidifies.31,38 If helium is held in a metastable liquid state, n0 is predicted31,38 to continue to decrease with increasing pressure, approximately exponentially, but is predicted to remain finite at high pressure (e.g. 150 bars). Values of n0 predicted31,38 at higher pressures differ by a factor of two. Of great interest, superfluidity,41–44 in solid helium has recently been reported. The mechanism is not understood, but may be associated with superflow involving

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(a)

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Fig. 2. Bose–Einstein condensate fraction, n0 (T), in liquid 4 He at saturated vapor pressure (SVP) (p ' 0). (a) LHS: n0 (T) in bulk liquid 4 He observed35 (solid dots) and calculated using path integral Monte Carlo (PIMC) and diffusion Monte Carlo (DMC) methods. 36–38 At T = 0 K the observed n0 (0) = 7.25±0.75%.35 (b) RHS: n0 (T) observed in liquid 4 He confined in Vycor.18 The solid symbols are n0 obtained using a simple one parameter model to analyze the data showing n0 is similar in Vycor and in the bulk helium. The open circles are n0 obtained using an accurate model with bulk data.

point defects45,46 or extended defects such as dislocations,47 but this requires a very high density of dislocations. The corresponding condensate fraction in crystalline solid helium is predicted to be small: (1) n0 ≤ 10−8 in a perfect crystal48,49 and (2) n0 ' 0.23 % in a crystal containing a 1 % concentration of vacancies.46 An n0 ' 0.5 % is predicted for solid helium held in an amorphous or glass state.48 Our recent measurement50 places n0 ≤ 1% at a pressure p = 41 bars in this rapidly developing field. The RHS of Fig. 2 shows the condensate fraction in liquid helium confined in Vycor at SVP. The neutron scattering data in Vycor are significantly less accurate than the data in bulk helium. For this reason the data must be analysed using a simple model that contains not more than one free parameter. If the Vycor and bulk data are both analysed using the same simple model, we obtain similar n0 values, the solid points in the LHS of Fig. 2. Thus n0 (T) in liquid helium in Vycor and in bulk are similar, within precision. The data are not precise enough to distinguish whether n0 (T) in Vycor goes to zero at 2.05 K (Tc in Vycor) or at Tλ = 2.17 K. The phonon-roton (P-R) energy dispersion curve in bulk superfluid helium at low temperature is shown on the LHS of Fig. 3. The P-R energy at higher wave vectors, Q, beyond the roton goes up to twice the roton energy, ∆, and is limited by 2∆ for Q ≥ 3 ˚ A−1 . If the energy exceeds 2∆, the single P-R mode can decay into two rotons. As Q increases beyond the roton region, less weight appears in the sharp P-R mode component of S(Q, ω) below (at) 2∆ and more lies above 2∆ in the form of broad response until the sharp P-R component disappears entirely at

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3

4

0.0 0.0

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Fig. 3. Phonon-roton energy dispersion curve at SVP (p ' 0). (a) LHS: bulk superfluid 4 He observed at low T , 5, Ref. 51, 4 Ref. 52. 24 is twice the roton energy. The P-R energy cannot exceed 24. (b) RHS: superfluid 4 He in Vycor and aerogel at SVP. The circles show layer modes.

The RHS of Fig. 3 shows the P-R energy dispersion curve in superfluid 4 He confined in Vycor at T = 0.5 K. In fully filled porous media, the P-R mode energy is the same as in the bulk.17,53–55 Liquid helium in porous media also supports layer modes at wavevectors in the roton region, modes that propagate in the liquid layers adjacent to the porous media walls (see Fig. 3 LHS and Refs. 53,54,56,57). In bulk liquid helium, sharply defined P-R modes at higher wave vectors exist only in the superfluid phase.58 As temperature is increased, the P-R mode in the superfluid broadens and the intensity in the modes decreases. The intensity in the P-R mode goes to zero at Tλ . This is particularly clear at wave vectors in the “maxon” (Q ' 1.1 ˚ A−159–64 and “beyond the roton” (Q ≥ 2.6 ˚ A−1 )52,65,66 regions. At these wave vectors, the sharp P-R mode is at relatively high energy (near 2∆). In the superfluid phase below Tλ , S(Q, ω) has three components, a low energy component arising from the Bose thermal factor, the P-R mode and broad intensity above 2∆. As the intensity in the P-R mode decreases with increasing T there is a compensating increase in the Bose thermal factor at lower energy - with the broad component above 2∆ changing little with temperature. In the normal phase the thermal intensity at low energy plus the broad component remains and is largely independent of temperature between Tλ and 4.2 K. The disappearance of the roton mode (a mode at lower energy) can also be seen60,63 if the resolution is sufficiently high67 to distinguish clearly the disappearance of the roton from the growing intensity at low energy. The shape of the response67 is quite different above and below Tλ . This temperature dependence may be related58 to BEC. When there is BEC in a Bose fluid, the single particle excitations and the density excitations (P-R mode) have the same energy.34,68–70 There are therefore no separate single particle excitations lying below the P-R mode to which the mode can decay. When there is

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BEC the P-R mode can decay only to other P-R modes. The P-R mode is uniquely sharp in superfluid 4 He at low temperature. In normal 4 He and liquid 3 He where there is no BEC, S(Q, ω) is many orders broader at higher Q as in other normal fluids. In bulk liquid helium, superfluidity, BEC and well defined P-R modes at higher Q all disappear at Tλ .

Q = 1.95 Å

expt. data Snor(Q,ω) SP-R(Q,ω) SL(Q,ω) S(Q,ω)

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(a) Fig. 4. Temperature dependence of the dynamic structure factor S(Q, ω) and weight f S (T ) in the phonon-roton mode of liquid 4 He confined in 44 ˚ A gelsil at SVP (p ' 0) where Tc = 1.92 K. (a) A−1 showing a mode at T = 1.90 K A−1 (roton) and Q = 1.4 ˚ S(Q, ω) at wavevectors Q = 1.95 ˚ and T = 2.00 K.17 (b) Fraction fS (T ) of the total S(Q, ω) taken up by the phonon-roton mode averaged over all P-R modes (all Q) versus temperature (solid circles). 17 The arrow indicates the Tc for 4 He in the present 44 ˚ A gelsil. A finite fS (T ) above Tc and therefore a P-R mode above Tc is indicated. Shown in the inset is fS (T ) for 4 He in Vycor14 where Tc = 2.05 K.

The LHS of Fig. 4 shows the temperature dependence of S(Q, ω) of liquid 4 He confined in 44 ˚ A mpd gelsil17 where15 Tc = 1.92 K. The P-R mode is identified by the dashed line. We see that there is an identifiable P-R mode at the wave vectors shown up to 2.00 K. The RHS of Fig. 4 shows the weight, fS (T ), in the P-R mode as a function of temperature (solid dots) averaged over many Q values.17 Weight in the P-R mode persists up to T = 2.10–2.15 K, close to Tλ . As found in Vycor,9,14,54 and a 25 A gelsil55 there is a P-R mode at temperatures above Tc , up to approximately Tλ . Since a well defined P-R mode is associated with BEC, this suggests that, in contrast to the bulk, there is BEC above Tc , at temperatures up to Tλ . This is interpreted9,14,17 as BEC that is not connected across the sample but rather is localized to specific regions in the porous media. The picture is islands of BEC having independent phases separated by normal fluid. There is no phase coherence and therefore no superflow across the sample. In this picture the superfluid to non-

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superfluid transition at Tc is associated with an extended to localized BEC cross over. In the temperature range, Tc ≤ T ≤ Tλ , there is localized BEC. 3. Confined Helium under Pressure The phase diagram of helium in 25 ˚ A gelsil depicting the superfluid phase and the liquid-solid (freezing and melting) transitions is shown in Fig. 5a.4,6,71 As p is increased, Tc decreases from Tc = 1.4 K. at SVP to Tc = 0 K at p = 34 bars.4 A ρS = 0 at T = 0 K indicates a possible Quantum Phase Transition (QPT) from the superfluid to normal liquid phase at a critical pressure pc = 34 bars. This is based on superfluid density measurements made using a torsional oscillator. 6 5

Pressure (MPa)

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Fig. 5. Phase diagrams of confined in gelsil. (a) LHS: in 25 ˚ A mpd gelsil showing superfluid,4 normal liquid and solid phases (from Shirahama Ref. 6). Solid line is bulk helium. A gelsil showing as a solid line and points the limits where P-R modes are (b) RHS: 4 He in 34 ˚ observed. The solid line is interpreted as marking the boundary of the existence of BEC in the liquid. The superfluid phase in 25 A gelsil observed by Yamamoto et al.4 is reproduced for clarity. 4 He

4 He

The classical expression for the enhancement of the freezing pressure ∆p above the bulk value when a liquid is confined in a spherical pore of diameter d is ∆p = 4γ/d, where γ is the solid/liquid interface surface tension. Since the gelsils have a broad distribution of pore diameters, δd ' d, this suggests a distribution of freezing pressures reflecting the distribution of pore diameters. We do not expect this expression to hold well in irregular and small pore media such as the present gelsils. However, it does suggest that freezing will take place first in the very largest pores and freezing beyond that will require a higher pressure. Direct simulations72,73 suggest freezing to an amorphous solid in small systems72 that takes place over a range of temperatures/pressures and layer by layer freezing in porous media. 73 The freezing pressure shown in Fig. 5a is the onset of freezing; that is, the initial freezing in the largest pores as the sample is cooled. The extrapolation of the freezing onset line to T = 0 K suggests an onset of freezing at T = 0 K at p = 34.2–36.5

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Fig. 6. Phonon-roton energy dispersion curve of superfluid at p = 31.2 bars in gelsil and roton energy versus pressure. (a) LHS: P-R dispersion curve at p = 31.2 bars in 34 ˚ A gelsil showing modes observed only at wavevectors 1.6 ≤ Q ≤ 2.3 ˚ A−1 . 24 is twice the roton energy. Modes in the maxon region (Q ∼ 1.0 ˚ A−1 ) where the P-R energy exceeds 24 are not observed. (b) RHS: roton energy 4 versus pressure in bulk (p ≤ 25.3 bars) and confined helium (p ≥ 25.3 bars) showing the decrease of 4 with p.

bars.6 Specifically, at 38.9 bars, Shirahama et al.6 observe onset of freezing at T = 1.2 K with a small pressure drop on freezing of δp ' 0.6 bars. Freezing at p = 41.5 bars and p = 49.0 bars shows a larger pressure drop on freezing, δp ' 1.7 bars and δp ' 3.0 bars, respectively, indicating that a larger fraction of the liquid is solidifying at the higher pressures. This is consistent with gradual freezing and a distribution of freezing pressures.

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0.04

ROTON

P = 25.4 bars P = 27.3 bars P = 31.2 bars P = 34.4 bars P = 37.8 bars P = 44.8 bars

SOLID PHONONS

0.2 0.15 0.1

0.02

0.05

0

0 0

0.5

1 E(meV)

(a)

1.5

2

0

0.5

1 E(meV)

1.5

2

(b)

Fig. 7. Pressure dependence of the dynamic structure factor S(Q, ω) of 4 He at low temperature confined in gelsil for wavevector Q = 2.1 ˚ A−1 (roton) showing the disappearance of the mode at p = 37.7 bars. (a) LHS: 25 ˚ A mean pore diameter gelsil. (b) RHS: 34 ˚ A mean pore diameter gelsil.

The aim of our neutron scattering measurements in 25, 34 and 44 ˚ A mpd gelsil at higher pressure is to determine the domain of existence of P-R modes in p and T to provide more information on the possible QPT. The present 25 ˚ A gelsil is

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the same as used by Yamamoto et al.4 to determine ρS and by Shirahama et al.6 and Yamamoto et al.71 to determine the freezing pressure. The 25 ˚ A gelsil sample was kindly provided by K. Shirahama and enables us to compare directly with their data. Our data for 44 ˚ A mpd gelsil has already appeared.15 For pressures up to 25.3 bars (the bulk liquid freezing pressure) we observe P-R modes at all wave vectors. The roton energy, ∆, decreases with increasing pressure as shown on the LHS of Fig. 6. The P-R energy cannot exceed 2∆ without decaying into two rotons. The “maxon” energy reaches 2∆ at p = 20 bars and remains at 2∆ between 20 and 25 bars. At pressures above 25.3 bars we no longer observe a P-R mode in the maxon region.15 We also do not observe a mode in the phonon region but this is probably because the intensity in the mode is low and our sample volume is small. At p = 31.2 bars and above we observe modes only in the wave vector range 1.6 ≤ Q ≤ 2.3 ˚ A−1 as shown in the RHS of Fig. 6. It is only in this limited wave vector range that the P-R mode energy lies below 2∆ at p = 31.2 bar. As pressure is increased further, the intensity in the remaining modes decreases and modes cease to be observed at all at p = 37.8 bars. Specifically, Fig. 7 shows the pressure dependence of S(Q, ω) at the roton wave vector in the 25 ˚ A and 34 ˚ A mpd gelsils. The low energy peak at ω ' 0.6 meV is the roton. The peak position (energy) of the roton decreases with increasing pressure. At pressures above 31.2 bars the intensity in the roton peak begins to drop and no peak is observed above 37.8 bars. The higher energy peak is phonon modes arising from the crystalline solid helium that lies between the gelsil and the cell walls. Note that a fresh sample is grown at each pressure so that the orientation of the solid helium crystals around the gelsil differs from pressure to pressure. The roton mode disappears at the same pressure in 25, 34 and 44 ˚ A mpd gelsils within precision. At p = 31.2 bars we observe the P-R roton mode up to 1.3 K only (see Fig. 5). We interpret the loss of P-R modes as signalling the loss of localized BEC. A domain of existence of localized BEC is shown on the RHS of Fig. 5. This interpretation should be valid for T ≥ 0.5–1.0 K. At lower temperature the P-R modes might also disappear because of solid formation. The pressures shown in Fig. 6 and quoted in our measurements are pressures at which the bulk solid helium freezes around the gelsil. This is at 2.0–2.1 K for 35–40 bars depending on the pressure. When the cell is cooled from the bulk freezing line to T ≤ 0.4 K, there will be a pressure drop of 1.0 - 1.5 bars in the cell. For example, beginning at 38.9 bars on the bulk melting line, Shirahama finds a pressure drop on cooling in his cell containing 25 ˚ A gelsil of 1.4 bars, with 0.6 bars of this arising from some solidification in the gelsil. Thus the pressure in our cell at which the roton is last observed at T ≤ 0.4 K is pB = 36.3 to 36.8 bars. Summarizing, the present results indicate: ˚ gelsil, a 1. A P-R mode exists up to a pressure pB = 36.3–36.8 bars in 25 A pressure greater than pc = 34 bars at which the superfluid vanishes. There is therefore sufficient liquid in the gelsil at pressures above pc to support a P-R

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mode. This supports the interpretation that the transition at pc is a Quantum Phase Transition in the fluid, from the superfluid to a “normal” liquid phase. 2. In the pressure range pc ≤ p ≤ pB , the liquid supports P-R modes but is not a superfluid. The existence of P-R modes suggests the existence of BEC. This BEC is interpreted as localized BEC, i.e. islands of BEC separated by regions of normal fluid as observed at SVP for Tc ≤ T ≤ Tλ . 3. At all pressures and temperatures, the superfluid phase in confinement appears to be surrounded by a localized BEC or glass phase. 4. The superfluid - normal transition in porous media is associated with an extended to localized BEC cross over. 5. From the phase diagrams in Fig. 5, the loss of P-R modes at temperatures above T = 0.5–1.0 K may be associated with the loss of BEC in the liquid phase. The loss of P-R modes at the lowest temperatures, T ≤ 0.5 K, could arise from loss of BEC, partial solidification in the gelsil or both. 6. A most interesting question is whether the loss of the P-R mode seen in 25, 34 and 44 ˚ A gelsil at pB = 36.3–36.8 bar is an intrinsic property of bulk helium under pressure or a result of confinement.

Acknowledgment ˚ mpd The authors are particularly grateful to K. Shirahama for supplying the 25 A gelsil sample and for valuable discussions. Support from the Institut Laue Langevin is greatly appreciated. This work was partially supported by the USDOE Grant No. DOE-FG02-03ER46038.

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TOPOLOGICAL QUANTUM ORDER: A NEW PARADIGM IN THE PHYSICS OF MATTER Z. NUSSINOV1 and G. ORTIZ2 ∗ 2 Department

of Physics, Washington University, St. Louis, MO 63160, USA

1 Department

of Physics, Indiana University, Bloomington, IN 47405, USA ∗ Contact: [email protected]

We prove sufficient conditions for Topological Quantum Order (TQO). The crux of the proof hinges on the existence of low-dimensional Gauge-Like Symmetries (GLSs), thus providing a unifying framework based on a symmetry principle. All known examples of TQO display GLSs. Other systems exhibiting such symmetries include Hamiltonians depicting orbital-dependent spin exchange and Jahn-Teller effects in transition metal orbital compounds, short-range frustrated Klein spin models, and p+ip superconducting arrays. We analyze the physical consequences of GLSs (including topological terms and charges) and, most importantly, show the insufficiency of the energy spectrum, (recently defined) entanglement entropy, maximal string correlators, and fractionalization in establishing TQO. Duality mappings illustrate that not withstanding the existence of spectral gaps, thermal fluctuations can impose restrictions on suggested TQO computing schemes. Our results allow us to go beyond standard topological field theories and engineer new systems with TQO. Keywords: Topological quantum order; fractionalization; non-local orders.

1. Introduction The Landau theory of phase transitions is a landmark in physics.1 Essential is an order parameter characterizing the thermodynamic phases of the system. A new paradigm, Topological Quantum Order (TQO),2 extends the Landau symmetrybreaking framework. At its core, TQO is intuitively associated with insensitivity to local perturbations. As such, TQO cannot be described by local order parameters. Interest is catalyzed by the prospect of fault-tolerant quantum computation.3 Several inter-related concepts are typically invoked in connection to TQO: symmetry, degeneracy, fractionalization of quantum numbers, maximal string correlations (non-local order), among others. The main issue is what is needed to have a system with TQO. But a real problem is that there is no unambiguous definition of TQO. The current article aims to show relations between these different concepts, by rigorously defining and establishing the equivalence between some of them and more lax relations amongst others. Most importantly, we (i) prove that systems harboring generalized d-dimensional (with d = 0, 1, 2) Gauge-Like Symmetries (d-

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GLSs) exhibit TQO; (ii) analyze the resulting conservation laws and the emergence of topological terms in the action of theories in high space dimensions; (iii) affirm that the structure of the energy spectrum is irrelevant for the existence of TQO (the devil is in the state itself); (iv) establish that, fractionalization, string correlators, and entanglement entropy are insufficient criteria for TQO; (v) report on a general algorithm for the construction of string correlators; (vi) suggest links between TQO and problems in graph theory. Our goal is to provide a unifying framework allowing the creation of new physical models displaying TQO. A very detailed review and derivation of the results presented here is available in [4]. We focus on quantum lattice systems (and their continuum extension) having QD Ns = µ=1 Lµ sites, with Lµ the number of sites along each space direction µ, and D the dimensionality of the lattice Λ. Associated to each lattice site (or mode, or bond, etc) i ∈ ZNs there is a Hilbert space Hi of finite dimension D. The Hilbert N space is the tensor product of the local state spaces, H = i Hi , in the case of distinguishable subsystems, or a proper subspace in the case of indistinguishable ones. Statements about local order, TQO, fractionalization, entanglement, etc., are relative to the particular decomposition used to describe the physical system. Typically, the most natural local language 5 is physically motivated. 2. Topological Quantum Order and Symmetries To determine what is needed for TQO, we start by defining it. Given a set of N orthonormal ground states (GSs) {|gα i}α=1,...,N and a (uniform) gap to excited states, TQO exists if for any bounded operator V with compact support (i.e. any quasi-local operator), hgα |V |gβ i = v δαβ + c,

(1)

where v is a constant and c is a correction that it is either zero or vanishes exponentially in the thermodynamic limit. We will also examine a finite temperature (T > 0) extension for the diagonal elements of Eq. (1), hV iα ≡ tr (ρα V ) = v + c

(independent of α),

(2)

with ρα = exp[−Hα /(kB T )] a density matrix corresponding to the Hamiltonian H endowed with an infinitesimal symmetry-breaking field favoring order in the state |gα i. A system displays finite-T TQO if it satisfies both Eqs. (1), and (2). A d-GLS of a theory given by H (or action S) is a group of symmetry transformations Gd such that the minimal non-empty set of fields φi changed by the group operations spans a d-dimensional subset C ⊂ Λ. These transformations can S Q be expressed as:6 Ulk = i∈Cl gik , where Cl denotes the subregion l, and Λ = l Cl (the extension of this definition to the continuum is straightforward). Gauge (local) symmetries correspond to d = 0, while in global symmetries the region influenced by the symmetry operation is d = D-dimensional. These symmetries may be Abelian or non-Abelian.

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3. Physical Consequences What are the physical consequences of having a system endowed with a symmetry group Gd ? Symmetries generally imply the existence of conservation laws and topological charges with associated continuity equations. We find that systems with d-GLSs lead to conservation laws within d-dimensional regions. To illustrate, con~ x) = sider the (continuum) Euclidean Lagrangian density of a complex field φ(~ P P 1 1 2 2 (φ1 (~x), φ2 (~x), φ3 (~x)) (~x = (x1 , x2 , x3 )): L = 2 µ |∂µ φµ | + 2 µ |∂τ φµ | + W (φµ ), P P with W (φµ ) = u( µ |φµ |2 )2 − 21 µ m2 (|φµ |2 ) and µ, ν = 1, 2, 3. L displays the continuous d = 1 symmetries φµ → eiψµ ({xν }ν6=µ ) φµ . The conserved d = 1 Noether currents are tensors given by jµν = i[φ∗µ ∂ν φµ − (∂ν φ∗µ )φµ ], which satisfy d = 1 conservation laws [∂ν jµν + ∂τ jµτ ] = 0 (with no summation over repeated indices implicit). What is special about d-GLSs is that there is a conservation law for each line associated with a fixed value of all coordinates xν6=µ relating to the topological R charge Qµ ({xν6=µ }) = dxµ jµτ (~x). Can we spontaneously break d-GLSs? The absolute values of quantities not invariant under Gd are bounded from above by the expectation values that they attain ¯ (or corresponding action S) ¯ which is globally inin a d-dimensional Hamiltonian H variant under Gd and preserves the range of the interactions of the original systems.6 As the expectation values of local observables vanish in low-d systems, this bound strictly forbids spontaneous symmetry breaking (SSB) of non-Gd invariant local quantities in systems with interactions of finite range and strength whenever d = 0 (Elitzur’s theorem), d = 1 for both discrete and continuous Gd , and (as a consequence of the Mermin–Wagner–Coleman theorem) whenever d = 2 for continuous symmetries.6 Discrete d = 2 symmetries may be broken (e.g. the finite-T transition of the D = 2 Ising model, and the d = 2 Ising GLS of D = 3 orbital compass systems). In the presence of a finite gap in a system with continuous d ≤ 2 symmetries, SSB is forbidden even at T = 0.6 The absence of SSB of non-GLSs invariant quantities is due to the presence of low-dimensional topological defects such as: domain walls/solitons in systems with d = 1 discrete symmetries, vortices in systems with d = 2 U (1) symmetries, hedgehogs for d = 2 SU (2) symmetries. Transitions and crossovers can only be discerned by quasi-local symmetry invariant quantities (e.g. Wilson-like loops) or, by probing global topological properties (e.g. percolation in lattice gauge theories7 ). Extending the bound of [6] to T = 0, we now find that if T = 0 SSB is precluded in systems with d-GLSs then it will also be precluded in the higher-dimensional system for quantities not invariant under exact or T = 0 emergent d-GLSs. Exact symmetries refer to [U, H] = 0; in emergent symmetries5 unitary operators U ∈ Gemergent are not bona fide symmetries ([U, H] 6= 0) yet become exact at low energies: when applied to any GS, the resultant state must also P reside in the GS manifold, U |gα i = β uαβ |gβ i. Thus, SSB of T = 0, d = 0 discrete or d ≤ 1 continuous symmetries, in systems of finite interaction strength and range, cannot occur. In gapped systems, T = 0 SSB of d ≤ 2 continuous symmetries is prohibited.

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Degeneracies imply the existence of symmetries which effect general unitary transformations within the degenerate manifold and act as the identity operator outside it. In this way, GS degeneracies act as exact GLSs. The existence of such unitary symmetry operators (generally a subset of SU (N )) allows for fractional charge (“N -ality” in the SU (3) terminology of quantum chromodynamics) implying that degeneracy allows for fractionalization defined by the center of the symmetry group. The (m-)rized Peierls chains constitute a typical example of a system with universal (m−independent) symmetry operators,4 where fractional charge quantized in units of e∗ = e/m with e the electronic charge is known to occur.8 The bounds above generalize to these symmetries. Different Peierls chain GSs break discrete d = 1 symmetries (violating Eq. (1) in this system with fractionalization). The Fermi number Nf in the Peierls chain and related Dirac-like theories is an integral over spectral functions;9 the fractional portion of Nf stems from soliton contributions and is invariant under local background deformations. When the bound of [6] is applied anew to correlators and spectral functions, it dictates the absence of quasiparticle (qp) excitations in many instances.10 Here we elaborate on this: the bound of [6] mandates that the absolute values of nonQ P symmetry invariant correlators |G| ≡ | Ωj aΩj h i∈Ωj φi i| with Ωj ⊂ Cj , and {aΩj } c-numbers, are bounded from above (and from below for G ≥ 0 (e.g. that corresponding to h|φ(k, ω)|2 i)) by absolute values of the same correlators |G| in a d-dimensional system defined by Cj . In particular, {aΩj } can be chosen to give the Fourier transformed pair-correlation functions. This leads to stringent bounds on viable qp weights and establishes the absence of qp excitations in many cases. In high-dimensional systems, retarded correlators G generally exhibit a resonant (qp) k contribution. Here, G = Gres (k, ω) + Gnon−res (k, ω) with Gres (k, ω) = ω−Zk +i0 +. In low-dimensional systems, the qp weight Zk → 0 and the poles of G are often replaced by weaker branch cut behavior. If the momentum k lies in a lower d-dimensional region Cj and if no qp resonant terms appear in the corresponding lower-dimensional spectral functions in the presence of non-symmetry breaking fields then the upper bound6 on the correlator |G| (and on related qp weights given by limω→k (ω − k )G(k, ω)) of non-symmetry invariant quantities mandates the absence of normal qps. Putting all of the pieces together we see that if fractionalization occurs in the lower-dimensional system then its higher-dimensional realization follows. Our central contention is that in all systems known to harbor TQO (and in new examples), d-GLSs are present. Old examples include: Quantum Hall systems, Z2 lattice gauge theories, the Toric-code model3 and other systems. In all cases of TQO, we may cast known “topological” symmetry operators as general low-dimensional d ≤ 2 GLSs (e.g. in the Toric code model, there are d = 1 symmetry operators spanning toric cycles). The presence of these symmetries allows for the existence of freely-propagating decoupled d-dimensional topological defects (or instantons in (d + 1) dimensions of Euclidean space time) which eradicate local order. These defects enforce TQO. We now state a central result:

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Theorem. When, in a system of finite interaction range and strength which satisfies Eq. (1), all GSs may be linked by discrete d ≤ 1 or by continuous d ≤ 2 GLSs U ∈ Gd , then the system displays finite-T TQO. Proof: Let us start by decomposing V = V0 + V⊥ . Here, V0 is the portion that transforms as a singlet under Gd i.e., [U, V0 ] = 0. To prove the finite-T relation of Eq. (2), we write the expectation values over a complete set of orthonormal states {|ai}, hV0 iα = lim

h→0+

= lim+ h→0

P

−β(Ea +φa α h) a ha|V0 |aie P −β(E +φa h) a α ae

P

−β(Eb +φbβ h) b hb|V0 |bie P −β(Eb +φb h) β be

= lim

h→0+

= hV0 iβ .

P

a ha|U

†

P

V0 U |aie

a

−β(Ea +φa h) U†β a

e−β(Ea +φU β h) (3)

Here, we invoked U |ai ≡ |bi, and Ea = Eb (as [U, H] = 0). The term φaα monitors the order parameter content of the state |ai vis-` a-vis that preferred by the GS |g α i. a b = φ , which is evident by the application of a In the above derivation, φaα = φU Uα β simultaneous unitary transformation in going from |ai →U |bi and |gα i →U |gβ i. V⊥ is not invariant under Gd ([U, V⊥ ] 6= 0) then by the theorem of [6], hV⊥ iα = 0, i.e. for systems with low-dimensional GLSs SSB is precluded. Eq. (3) is valid whenever [U, V0 ] = 0 for any symmetry U . However, [U, V⊥ ] 6= 0 implies hV⊥ iα = 0 only if U is a low-dimensional GLS. In systems in which not all GS pairs can be linked by the use of low-dimensional GLSs U ∈ Gd (U |gα i = |gβ i), finite-T SSB may occur. We conclude this proof with two remarks:4 (i) T = 0 TQO holds whenever all GSs may be linked by continuous d ≤ 2 GLSs. (ii) In many systems (whether gapped or gapless), T = 0 TQO states may be constructed by employing Wigner–Eckart type selection rules for GLSs. A related consequence of systems with d-GLSs is that topological terms which appear in d + 1-dimensional theories also appear in higher D + 1-dimensional systems (D > d). These topological terms appear in actions S¯ which bound quantities not invariant under the d-GLSs. For instance, in the isotropic D = 2 (or 2+1) dimensional general spin t2g KK model,4,11 of exchange constant J > h0, the correR 1 ˆ 2− dxdτ v1s (∂τ m) sponding continuum Euclidean action is of the 1+1 form S¯ = 2g i θ vs (∂x m) ˆ 2 + iθQ + Str , Q = 8π µν · (∂µ m ˆ · ∂ν m), ˆ with µ, ν ∈ {x, τ } and νµ the rank two Levi Civita symbol. Here, as in the non-linear-σ model of a spin-S chain, m ˆ a normalized slowly varying staggered field, g = 2/S, vs = 2JS, θ = 2πS, and Str a “transverse-field” action term which does not act on the spin degrees of freedom along a given chain.6 Q is the Pontryagin index corresponding to the mapping between the 1+1 dimensional space-time (x, τ ) plane and the two-sphere on which m ˆ resides. This 1+1 dimensional topological term appears in the 2+1 dimensional KK system even for arbitrary large positive coupling J. This, in turn, places bounds on the spin correlations and implies, for instance, that in D = 2 integer-spin t2g KK systems, a finite correlation length exists.

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4. What Characterizes Topological Quantum Order? We will see that neither the spectrum of H, nor the entanglement entropy,12 nor string correlations are sufficient criteria. That the spectrum, on its own, is insufficient is established by counter-examples, e.g. that of the D = 3 Z2 lattice gauge theory which is dual to a D = 3 Ising model.13 Albeit sharing the same energy spectrum, the gauge theory displays TQO while the Ising model harbors a local order. A spectral equivalence exists between other TQO inequivalent system pairs. For example, Kitaev’s3 and Wen’s models14 are equivalent and have a spectrum which is identical to that of an Ising spin chain with nearest neighbor interactions.4 The mappings between these systems to an Ising chain demonstrate that despite the spectral gap in these systems, the toric operator expectation values may vanish once thermal fluctuations are present. These mappings also illustrate that the quantum states themselves in a particular (operator language) representation encode TQO and that the duality mappings, being non-local in the original representation, disentangle the order. Thus, we cannot, as coined by Kac, “hear the shape of a drum”. The information is in the eigenvectors. It was recently suggested12 that a non-local borne deviation γ > 0 from an asymptotic area law scaling for the entanglement entropy constitutes a sharp measure of TQO. We find that on its own, this (defined) measure of “Topological Entanglement Entropy” (TEE) — a lnear combination of entanglement entropies which is meant to extract γ — might not always be a clear marker of TQO. An example is afforded by Klein spin models whose GS basis is spanned by singlet product states. In most of these GSs, local measurements can lead to different expectation values (and thus do not satisfy Eq. (1)). Here, we find4 (i) an entanglement entropy which deviates from an area law within the set of all GSs which do not host TQO. Such an arbitrary — contour-shape dependent — finite TEE in a non-TQO system is clearly not in accord with the conjecture of [12]. Moreover, (ii) a subset of TQO GSs of the Klein model constructed as uniform linear superpositions of the singlet product states may have zero entanglement entropy [γ = 0]. This suggests that the criterion of [12] adducing TQO from a finite γ can be violated and may require additional improvements, such as the specification of an exact limit and/or average for large contours. We conjecture that at least a demand that the TEE is not only finite but also contour independent needs to be amended. The insufficiencies of the spectra and entropy in determining if TQO is present have counterparts in the topology of graphs and in the Graph Equivalence Problem (GEP) in particular. The adjacency matrix of a graph has elements Cij = 1 if vertices i and j are linked by an edge and Cij = 0 otherwise. Vertex relabeling i → p(i) leaves a graph invariant but changes the adjacency matrix C according to C → C 0 = P † CP with P an orthogonal matrix which represents the permutation: P = δj,p(i) . The GEP is the following:15 “Given C and C 0 , can we decide if both correspond to the same topological graph?” The spectra of C and C 0 are insufficient criteria. Entropic measures15 are useful but also do not suffice.

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In many systems (with or without TQO), there are non-local “string” correlators which display enhanced (or maximal) correlations vis a vis standard 2-point correlation functions. We now outline an algorithm for the construction of such non-local correlators (in systems with uniform global order already present in their GS, the algorithm leads to the usual 2-point correlators). We seek a unitary transformation Us which rotates the GSs into a new set of states which have greater correlations as measured by a set of local operators {Vi }. These new states may have an appropriately defined polarization (eigenvalues {vi }) of either (i) more slowly decaying (algebraic or other) correlations, (ii) a uniform sign (partial polarization) or (iii) maximal expectation values vi = vmax for all i (maximal polarization). Cases (i) or (ii) may lead to a lower dimensional gauge like structure for the enhanced correlator. In systems with entangled GSs (such as in many (yet not all) GSs hosting Q T = 0 TQO), Us cannot be a uniform product of locals; Us 6= i∈Λ Oi . To provide a concise known example where these concepts become clear, we focus below on case (ii) within the well-studied AKLT Hamiltonian.16 Here, there is a non-trivial Q unitary operator Us ≡ j hgα |Vi Vj |gα i ≡ Gij . site out of the three S = 1 states), G ˜ ij becomes a non-local “string” correlator G ˜ ij = When Us is written in full, G Q |i−j| z 2 z z (−1) hgα |Si i j. The operators linking the GSs form a discrete (Z2 × Z2 ) global symmetry group16 and, as such, T = 0 SSB may occur. The general polarization operators are intimately tied to selection rules which hold for all of the GSs; these selection rules (and a particular low-energy projection of truncated Hamiltonians for gapped systems) enable the construction of general string and higher dimensional z |gα i “brane” correlators. In the AKLT problem, the local expectation value hg α |Si=1 depends on |gα i, violating Eq. (1) and suggesting that the AKLT chain is not topo˜ ij = 1) string correlators; here, logically ordered. We may also generate maximal (G the number of states with maximal polarization is finite and a non-local gauge-like structure emerges. Generally, in any system with known (or engineered) GSs, we may explicitly construct polarizing transformations Us .

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5. Conclusion To conclude, we proved a theorem establishing how d ≤ 2 dimensional GLSs in gapped systems can mandate TQO via freely propagating low-dimensional topological defects (e.g. d = 1 solitons). This result extends the set of known systems which exhibit TQO to include new orbital, spin, and Josephson junction array problems. All known TQO systems display GLSs.4 We examined physical consequences of GLSs (such as conservation laws and topological terms in high-D theories), and low-dimensional gauge-like structures in theories with entangled GSs in non-maximal string correlators (irrespective of TQO). We established insufficiencies of the Hamiltonian spectra, fractionalization, entanglement entropy, and maximal string correlations in determining TQO. References 1. L.D. Landau and E.M. Lifshitz, Statistical Physics (Butterworth-Heinemann, Oxford, 1980). 2. X-G. Wen, Quantum Field Theory of Many-Body Systems (Oxford University Press, Oxford, 2004). 3. A. Kitaev, Ann. Phys. 303, 2 (2003). 4. Z. Nussinov and G. Ortiz, cond-mat/0702377. 5. C. D. Batista and G. Ortiz, Adv. in Phys. 53, 2 (2004). 6. C. D. Batista and Z. Nussinov, Phys. Rev. B 72, 045137 (2005). 7. Z. Nussinov, Phys. Rev. D 72, 054509 (2005). 8. W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. Lett. 42, 1698 (1979). 9. A. J. Niemi and G. W. Semenoff, Phys. Rep. 135, 99 (1986). 10. Z. Nussinov, C. D. Batista, and E. Fradkin, cond-mat/0602569. 11. K. I. Kugel and D. I. Khomskii, Sov. Phys. JETP 37, 725 (1973). 12. A. Kitaev and J. Preskill, Phys. Rev. Lett. 96, 110404 (2006); M. Levin and X-G. Wen, Phys. Rev. Lett. 96, 110405 (2006). 13. F. J. Wegner, J. of Math. Phys. 12, 2259 (1971). 14. X-G. Wen, Phys. Rev. Lett. 90, 016803 (2003). 15. C. D. Godsil, D. A. Holton, and B. D. McKay, The spectrum of a graph, in Lecture Notes in Math. 622, 91 (Springer-Verlag, Berlin, 1977). 16. I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Commun. Math. Phys. 115, 477 (1988); T. Kennedy and H. Tasaki, Commun. Math. Phys. 147, 431 (1992).

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THERMAL RECTIFICATION IN ONE-DIMENSIONAL CHAINS Nan ZENG Jian-Sheng WANG Center for computational science and engineering, and Department of Physics, National University of Singapore, Singapore 117542, Republic of Singapore We studied the thermal rectification effect in one-dimensional chains using the nonequilibrium Green’s function method. Thermal rectification effect is observed in the presence of structure asymmetry and nonlinear interactions. We found that phonons with higher frequencies play a major role in thermal rectification. By controlling the phonon frequency band in the junction region, we are able to obtain rectification rate up to four percent. The results are comparable to experiment, and provide a clear interpretation of the physical processes involved. Keywords: Thermal transport; nanoscale; rectification; nonlinear; non-equilibrium green’s function.

1. Introduction Electronic diode, which is used to control current flow, has been invented for more than a century and plays an indispensable role in the everyday life. However, up to now, no counterpart is available to control thermal transport. A solid state thermal rectifier that can control heat flux would be both theoretically interesting and practically valuable. Theoretical attempts looking for special structures that will finally lead to thermal rectification have been actively going on in recent years. Terraneo et al.1 studied a one-dimensional chain model with on-site nonlinear interactions. By varying the heat bath temperatures on both sides of the chain, they suggested a possible implementation of thermal rectifier and explained in a band theory. A different model which concentrated on the interface effect was introduced by Li et al., 2 where two separate parts of nonlinear lattices were joined together with a harmonic chain. A sinusoidal on-site potential was used to represent the nonlinear interaction in the lattice. These studies achieved the rectification effect by manipulating the heat bath, while in reality, it is much more desirable to obtain the same effect by operating on the central junction part, which has been worked on in some recent studies.3,4 Until recently, most of the studies on thermal rectification have been carried out classically using the molecular dynamics method, except for a few papers 5–7 discussing the rectification of a simple two-level system quantum mechanically. In this paper, we studied the thermal conductance through a one-dimensional

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chain with monotonic changing mass using non-equilibrium Green’s function method. This first-principles method enables us to study the effective transmission probability of phonons with different frequencies, which is essential for understanding the thermal rectification effect. Our results clearly show how asymmetry of the structure and nonlinear interactions may result in thermal rectification. 2. Method Our model is a one-dimensional system which is separated into three parts. The right and left parts are semi-infinite linear leads which are acting as heat baths. The central junction part is coupled to the leads through linear interactions. It can have any structure and it also includes nonlinear interactions, which are our major interest. The Hamiltonian of the system is written as X H= Hα + uTL VLC uC + uTC VCR uR + Hn α=L,C,R

Hα =

X1 i

Hn =

2

u˙ 2αi +

X 1 kα,ij (uαi − uαj )2 2 i,j=i±1

(1)

X 1 X 1 k3 (ui − uj )3 + k4 (ui − uj )4 3 4 i,j=i±1 i,j=i±1

√ where ui = xi mi is the normalized displacement for atom i. Hα is the harmonic Hamiltonian of each part, VLC and VRC are the interactions between the junction and the lead, and Hn is the nonlinear interaction in the junction. In the ballistic electronic transport, the conductance can be calculated by the Landauer formula. In the thermal transport with nonlinear interactions, we can get a similar formula Z ∞ ∂ f (ω) 1 ωT [ω] dω, (2) κ= 2π 0 ∂T where T [ω] is an effective transmission function and represents the probability of a phonon with frequency ω to transmit from one side of the junction to the other side, and f (ω) is the Bose–Einstein distribution of the phonons in the heat bath. An explicit form of T [ω] is given in Ref. 9. 3. Results and Discussion Our model is a chain of atoms which only interact with their nearest neighbors. The interaction coefficients between adjacent atoms are obtained √ −1 from Morse function 2 2 D(exp(a(r − r0 ) ) − 1) , where D = 70kcal/mol, a = 5˚ A , and r0 is the equilibrium distance between adjacent atoms. The values of these parameters are obtained from Ref. 10. In our calculation, the third and forth order coefficients are needed, which are obtained by Taylor expansion of the potential around the equilibrium distance. The left lead and the right lead maintain a temperature difference ∆T which gives TL = T + ∆T /2 and TR = T − ∆T /2 for a particular temperature T ,

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or in reverse order. In the following part, without specification, the unit of mass is g/mol, the units of the second, third and fourth order interaction coefficients are eV/˚ A2 , eV/˚ A3 and eV/˚ A4 respectively, and the unit of temperature is K as usual. Figure 1 shows the transmission functions with two different heat bath setups. The junction part contains 20 atoms with a mass gradient of 0.2. The atoms in the left lead of both setups have mass ml = 12. The solid and plus lines have mass mr = 12 in the right lead, while the segmented and circled lines have mass mr = 57.6. For each configuration, the temperature of the heat baths can be swapped, thus there are four curves altogether. The temperature difference between the heat bath is ∆T = 240. The inlet represents the changing of the phonon frequency band in the chain for these setups. The transmission graph clearly shows that the frequency band is separated into two parts, a low frequency part and a high frequency part, which is marked with a vertical dotted line. We know that for a one-dimensional harmonic chain, the maximum pfrequency is determined by the interaction coefficient K with the relation ωmax = 4K/m, where m is the mass of the individual atom. In our calculation, with K = 17.3 and m = 12, the maximum frequency of the lead is 3.04, while the junction’s maximum mass is 57.6, it limits the maximum transmittable frequency to around 1.39. The conductance for heat flow from left to right is 0.148nW/K, and the conductance from right to left is 0.154nW/K. The rectification ratio α can be calculated as α=

|κL→R − κR→L | × 100%, κL→R

(3)

which gives 4% rectification. From the graph we notice that, at low frequency region, the curves for each setup match exactly, while in the high frequency region, the difference is obvious. The heat bath does not affect the rectification. It is totally determined by the junction region. When the narrower part of the phonon band is connected to a lower temperature heat bath, the phonon transmission is smaller, as high frequency phonons are prohibited to cross the junction region through collisions. On the other hand, when the heat baths are reversed, some lower energy phonons are converted to higher energy phonons, which fill the higher part of phonon band. This is how thermal rectification is achieved. 4. Conclusion In conclusion, we have discussed the thermal rectification effect of a one-dimensional chain with varying mass using the non-equilibrium Green’s function method. It is found that the asymmetry of the band structure and nonlinear interactions are essential for thermal rectification. Acknowledgment The authors thank Jingtao L¨ u and Jian Wang for discussions. This work is supported in part by a Faculty Research Grant.

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1 TL > TR, mr = 12 TL < TR, mr = 12 TL > TR, mr = 57.6 TL < TR, mr = 57.6

0.8

0.6

TL

TR

T(ω)

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0.4

0.2

0

0.5

1

1.5

ω

2

2.5

3

Fig. 1. Heat bath effect on thermal conductance. The solid and plus lines represent a right lead with mass mr = 12. The segmented and circled lines represent a right lead with mass m r = 57.6.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

M. Terraneo, M. Peyrard, and G. Casati, Phys. Rev. Lett. 88, 094302 (2002). B. Li, L. Wang, and G. Casati, Phys. Rev. Lett. 93, 184301 (2004). N. Yang, N. Li, L. Wang, and B. Li, Phys. Rev. B 76, 020301 (2007). G. Wu and B. Li, Phys. Rev. B 76, 085424 (2007). D. Segal and A. Nitzan, Phys. Rev. Lett. 94, 034301 (2005a). D. Segal and A. Nitzan, J. Chem. Phys. 122, 194704 (2005b). D. Segal, Phys. Rev. B 73, 205415 (2006). C. W. Chang, D. Okawa, A. Majumdar, and A. Zettl, Science 314, 1121 (2006). J.-S. Wang, N. Zeng, J. Wang, and C. K. Gan, Phys. Rev. E 75, 061128 (2007). S. L. Mayo, B. D. Olafson, and W. A. G. III, J. Phys. Chem. 94, 8897 (1990).

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AUTHOR INDEX Ac´ın A., 398 Aichhorn M., 336 Allmaier H., 336 Arias de Saavedra F., 152 Arrigoni E., 336 Astrakharchik G. E., 94, 116, 245 Baer M., 374 Bajdich M., 193 Barrett B. R., 172 Baym G., 65 Bisconti C., 152 Bishop R. F., 265 Boronat J., 116, 245, 312 Bossy J., 411 Bousquet G., 374 Buend´ıa E., 364 Caffarel M., 353 Campbell C. E., 39 Capuzzi P., 111, 120 Casulleras J., 116, 245, 312 Cataldo H. M., 120 Ceperley D. M., 217 Chin S. A., 203 Chioncel L., 336 Cirac J. I., 106, 398 Clark J. W., 3 Co’ G., 152 Colletti L., 213 Darradi R., 265 de Souza Cruz F. F., 379 Dean D. J., 168 Delaney K. T., 217 Deng S., 387 Dickhoff W. H., 148 Dinh P. M., 374 Fantoni S., 23 Federici F., 111

Fehrer F., 374 Finelli P., 176 G´ alvez F. J., 364 Galli D. E., 251, 300 Gandolfi S., 23 Gernoth K. A., 316 Glyde H. R., 411 Gordillo M. C., 312 Guardiola R., 281 Hagen G., 168 Hanke W., 336 Harrison M. J., 316 Hern´ andez E. S., 291 Hernando A., 291 Hjorth-Jensen M., 168 Holzmann M., 217 Horstmann B., 106 Hoyos J. A., 235 Jezek D. M., 120 Kanzawa H., 181 Kol´ aˇcek J., 332 Krotscheck E., 39 Kurbakov I. L., 245 Kusmartsev F. V., 323 Lapolli E. L., 379 Lewenstein M., 79, 398 Li P. H. Y., 265 Lipavsk´ y P., 332 Lipparini E., 213 Lisetskiy A. F., 172 Lozovik Yu. E., 245 M¨ uther H., 275 Maldonado P., 364 Martínez-Pinedo G., 156 Mayol R., 291

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Author Index

Mazzanti F., 116 Menotti C., 79 Messud J., 374 Mitas L., 193 Morales M. A., 217 Morawetz K., 332 Navarro J., 281 Navr` atil P., 172 Nesterenko V. O., 379 Nussinov Z., 423 Ortiz G., 20, 387, 423 Oyamatsu K., 181 Papenbrock T., 168 Papoular D., 94 Pearce J. V., 411 Pederiva F., 23, 213 Petrov D. S, 94 Pi M., 291 Pieri P., 75 Pierleoni C., 217 Polls A., 11, 275 Ramírez-Solís A., 353 Ramos A., 127, 275 Reatto L., 251, 300 Reinhard P.-G., 374, 379 Richter J., 265 Rios A., 275 Ritsch H., 346 Roscilde T., 106 Rossi M., 300 Rota R., 300

Saarela M., 16 Salomon C., 94 Sarsa A., 364 Schmidt K. E., 23 Schober H., 411 Sedrakian A., 138 Shlyapnikov G. V., 94 Sobnack M. B., 323 Spuntarelli A., 75 Stetcu I., 172 Strinati G. C., 75 Sumiyoshi K., 181 Suraud E., 374 Takano M., 181, 185 Tanaka K., 185 Tosi M. P., 111 Umrigar C. J., 213 Vary J. P., 172 Verstraete F., 53 Viola L., 387 Vitali E., 251, 300 Vojta T., 235 Walet N. R., 255 Wang J.-S., 431 Whaley K. B., 295 Wu W. M., 323 Zeng N., 431 Zillich R. E., 295 Zoubi H., 346

recent

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SUBJECT INDEX ab-initio calculations, 11, 60, 157, 170, 217, 219, 301, 342, 354

dispersive optical model, 148 drip lines, 149, 163

Anderson localization, 108 Andreev–Saint James states, 77

electronic structure, 199, 342, 360, 370 entanglement, 22, 53, 390, 399, 428 generalized, 388 Euler–Lagrange equation, 113, 182, 185, 296 exciton, 346

BEC-BCS crossover, 69, 75, 141, 255 Bogoliubov-de Gennes equations, 76 Born–Oppenheimer approximation, 95, 218, 281, 355 Bose–Einstein condensation, 66, 82, 111, 120, 139, 301, 379, 380, 411, 413 rotating, 203 Bose–Hubbard model, 84, 107, 346, 387 Brueckner–Hartree–Fock, 26, 275 carbon nanotubes, 312 charge density, 214 chiral symmetry, 70 theory, 133, 157, 169, 176 clusters, 158, 236, 282, 295, 338, 380 magical, 284 metal, 374 concurrence, 388, 400 correlated basis functions, 9, 12, 16, 24, 295 correlated density, 334 coupled cluster method, 168, 265, 267, 358 critical exponent, 236, 393, 412 cross section, 160 crystal structure bcc, 224 fcc, 224, 316 hcp, 304, 316 density functional theory, 18, 176, 354 dipolar gases, 80, 245 dipole mode, 214 disorder, 106, 226, 244, 265, 301, 411

Feenberg, 3, 17, 39 Fermi hypperneted chain, 11, 16, 32 mixtures, 95 superfluid, 65 transition, 160 fermion nodes, 194 ferromagnets, 239, 338, 392 Feshbach resonance, 69, 80, 103 Fock state, 107 fourth order algorithm, 203 fractal, 236 fractional quantum Hall effect, 13, 88 fractionalization, 423 Gamow–Hartree–Fock single-particle basis, 169 Gamow–Teller transition, 160 grain boundaries, 301 Green’s function, 13, 39, 139, 151, 256, 275, 339 Gross–Pitaevskii equation, 77, 112, 121, 205 harmonic approximation in solids, 99 Hartree–Fock, 148, 196, 222, 358, 360, 365 Hartree–Fock–Bogoliubov, 164 Heisenberg model, 60, 237, 267 Helium droplet, 17, 295

April 25, 2008

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recent

Subject Index

liquid, 71, 292, 312, 327, 413 solid, 301, 319 Hubbard model, 60, 338 Hubbard–Stratonovich transformation, 27, 256 hypernuclei, 127 hyperon-nucleon interaction, 128 Ising model, 60, 237, 392, 425 Jordan–Wigner diagonalization, 108, 391 Josephson effect, 13, 75 kaonic atoms, 132 nuclear states, 128 Kohn–Sham density functional theory, 176, 223 time-dependent Hartree–Fock method, 380 Landau critical velocity, 77 lowest level, 88 theory of phase transitions, 423 Laughlin state, 13, 88 Lennard–Jones interaction, 288, 375 local density approximation, 223, 342 time dependent, 375 maxon, 82, 415 Meissner effect, 72, 323 microscopic theory, 4, 12, 17 momentum distribution, 12, 24, 116, 154, 276 Monte Carlo, 109, 117, 193 auxiliary diffusion, 13, 26 coupled electron-ion, 217 diffusion, 97, 213, 246, 285, 295, 297, 313, 355, 367 fixed node, 219, 357, 367 fixed phase approximation, 28 Fourier path integral, 317 Green’s function, 26, 157, 368 path integral ground state, 301 variational, 246, 366 Mott insulator, 84, 108, 346, 387 muonic helium, 40 Néel state, 266

neutron matter, 33, 139, 155, 178, 183, 185 neutron star, 30, 72, 132, 144, 159, 181 nodal hypersurface, 194 nonlocal kinetic theory, 335 Nozières–Schmitt-Rink conjecture, 142 nuclear astrophysics, 24, 156 matter, 29, 67, 128, 138, 149, 153, 177, 181, 188, 263, 276 structure, 23, 158, 173, 176 nucleon-nucleon correlations, 149, 158 nucleosynthesis, 138, 163 one-body density matrix, 247, 300, 317 optical lattices, 53, 84, 94, 106, 348 ortho-Hydrogen, 282 pair distribution function, 45, 153, 185, 247, 296, 318 pairing, 13, 72, 82, 95, 139, 151, 179, 255, 339, 367 para-Hydrogen, 282 percolation, 235 entanglement, 399 pfaffian, 26, 198 phase imprinting, 121 phase transition, 406 classical, 267 dynamic, 394 first order, 72, 95, 218, 277 liquid-gas, 277 quantum, 20, 103, 235, 247, 266, 387, 417 second order, 72, 272 phonon-roton spectrum, 40, 51, 297, 411 plasmon, 144, 214, 222, 374, 379 polariton, 348 polarizability, 214, 375 polynomial trapping potential, 121 population transfer, 379 projected entangled pair states, 55 quantum communication, 22, 399 quantum dots, 213 quantum magnet, 235, 239, 265 quark, 130, 177 quark-gluon plasmas, 65 Raman scattering, 282, 380

April 25, 2008

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recent

Subject Index

439

rare gas materials, 375 rectification, 431 renormalization, 22, 53, 241, 256, 266, 343, 388 Renyi entropy, 59 ripplon, 297 rotational spectrum, 295 roton, 39, 82, 415

tensor correlations, 28, 158, 187 thermal transport, 431 Thomas–Fermi theory, 66, 121 three-nucleon interaction, 34, 157, 169, 172, 177, 182 topological quantum order, 55, 424 trimers, 101, 144 triple point, 272, 273

shell model, 5, 164, 172 single operator chain, 153 sound wave, 111 spin model J1 –J10 –J2 , 267 spin-1, 269 spin-1/2, 269, 391 stimulated Raman adiabatic passage, 380 strangeness nuclear physics, 127 stripe state, 270 subnuclear matter, 138 sum rules, 145, 153, 177, 214 superconductors, 72, 198, 323, 334, 390 high-temperature, 266, 336 supernovae, 67, 138, 156, 181 supersolid, 84, 100, 248, 254, 301, 413 symmetry breaking, 88, 176, 253, 339, 425

ultrarelativistic heavy ion collisions, 68 uniform limit approximation, 47, 49 unitarity regime, 69, 77 vacancies, 246, 301, 414 Van der Waals forces, 80, 283, 353, 375 variational cluster approach, 339 viscosity, 71 von Neumann entropy, 54 vortices, 70, 88, 111, 120, 203, 323, 335, 425 vycor, 414 Wigner crystal, 90

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